Maher's Digital World
Off Topic Discussion => Chit Chat => Topic started by: scarface on June 16, 2013, 03:58 PM

Clearly, a section mathematics is missing on this forum. Some readers are probably not interested only in downloading, even though I fear too many are coming only to download and leave.
So I have a little problem: Here is a random variable X and its distribution probability:
P(X=0)=1/4, P(X=1)=1/12 P(X=2)=1/6 P(X=3)=1/2
Its Expected value is E(X)=23/12
Im looking for the standard deviation. I found 0,909 but a program gives me 1.26. I recall the standard deviation is given by sqrt(E(X²)E(X)²) according to the formula of konig huygens.
It was an error, doing it again I found 1.26. thats ok.

Here is an interesting biquadriatic equation,that some of you can eventually resolve:
x^4(e²+1)x²+e²=0
tips: a change of variable is required. There are 2 couples of solutions.

I'm glad you opened this topic. Many years ago when I was in school I was a whiz at math. Unfortunately it's been many years and I didn't get the opportunity to study higher math. I still remember quite a bit.
In this example you gave, naturally the idea is to factor the equation. Hmm... I'm familiar with regular quadratic equations but not biquadratics. Tell me more.

Indeed,without e^2 at the end it coule be a factorisation. Its in fact a change of variable thats required: x^2=X. Then try to resolve X^2(e^2+1)X+e^2=0. Ill give the solution later.

here the solution:
So there is a classic polynomial to resolve: X²(e²+1)X+e²=0
Delta=(e²+1)²4e² and cant be used easily...But there is 1 obvious root: 1
We also know X2=c/X1 So We have the solutions:
X1=1, X2=e²/1=e²
Then we have to change the variable again to come back to the first equation:
as x²=X, x=sqrtX or x=sqrtX
then x=sqrtX1=1 or x=sqrtX2=e or x=sqrtX1=1 or x=sqrtX2=e
So the equation x^4(e²+1)x²+e²=0 has four solutions: {(1,e,1,e)}.

Here is a harder equation to resolve: (cosx)^3(sinx)^3=0
Here is the function f(x)=(cosx)^3(sinx)^3
(https://i.ibb.co/554jgnq/cosx3sinx3.jpg)

Here is the solution:
We are going to study the function h(x)=(cosx)^3(sinx)^3
h(x)=gof(x) with g(x)=x^3 and f(x)=cosxsinx.
We know that x^3 is bijective and therefore injective and
by definition x^3=x'^3<=>x=x'.
Then (cosx)^3=(cosx')^3<=>cosx=cosx'
and (sinx)^3=(sinx')^3<=>sinx=sinx'
then (cosx)^3(sinx)^3=0 <=> cosxsinx=0 <=> cosx=sinx
We know that cosx and sinx are 2pi periodic
and that cosx=sin pi/2 x
then cosx=sinx <=> sin pi/2x=sinx+2kpi with k a relative number
by identification pi/2x=x+2kpi <=> 2x=pi/2+2kpi <=> x=pi/4+kpi
So there is an infinity of solutions with a pi periodicity.

Probably it was a little hard. For the second exercice,i had the correction and for the third exercice i knew the method. Note that its essential to study the periodicity before the identification for any equation with cos, sin or tan. Clearly i dont deserve a field medal, the equivalent of nobels for mathematics. Im going to propose an exercise with vectors, the specialist humbert will appreciate.
If you have any problem of mathematics, feel free to ask.

So I have renamed the topic maths exercices. Its better like that. Im now proposing an exercice with vectors. I dont have the correction, but its probably quite easy, a drunk college student could probably resolve it. the wording was in french so I translated it:
The space is linked to an orthonormal system (O; i; j). S is a Real number
We give the points A (8 ; 0 ; 8 ), B (10 ; 3 ; 10) and the line (D) defined by the parametric equations:
x = − 5 + 3 s
y = 1 + 2 s
z = − 2 s s= IR
1) a) give a system of parametric equations defined by the line Δ defined by A and B.
b) Demonstrate that (D) and (Δ) are not coplanar.
2) a) The plane (P) is parallel to (D) and contains (Δ). Demonstrate the vector n (2 ; − 2 ; 1) is normal to (P) then determine an cartesian equation of (P) .
b) Demonstrate that that the distance of any point M of (D) to (P) is independant of M.
c) give a system of parametric equations of the line defined by the intersection of ( P) with the plane (xOy).
3) the sphere (S) is tangent to the plane ( P) on the point C(10 ; 1 ; 6). The center Ω of S is at the distance d = 6 of ( P) , on the same side that O. Give the cartesian equation of S .

So Im going to give a correction for the last exercice. Humbert has already revised its factorisation and is perhaps waiting impatiently to check his results. I did this one under notepad to copy it directly, but I couldn't do the last part which is difficult.
1)
a) The vector AB(2;3;2) is a direction vector of Δ.
Then,
x = 8+2t
y = 3t t=IR
z = 8+2t
b)2 lines are coplanar if they are not parallel, nor secant.
let's take the vector u(3,2,2) a direction vector of (D).
lets prove they are not parallel:
AB(2,3,2) and u(3,2,2) are not collinear (there is no real number k such as AB=ku)
then (Delta) and (D) are not parallel.
lets prove they are not secant:
We have to resolve the system to eventually find a point of intersection:
x =  5 + 3 s
y = 1 + 2 s
z =  2 s
x = 8+2t
y = 3t
z = 8+2t
<=>
5+3s = 2s <=> s= 1
y = 1 + 2 s
z =  2 s
x = z
y = 3t
z = 8+2t
<=>
s = 1
y = 3
z = 2
y = 3 <=> t = 1
z = 10
The system has no solution then the lines (AB) and Δ are not secant.
The lines (AB) and Δ are not parallel nor secant, they are not coplanar.
2)
a) The plane (P) is parallel to (D) and contains Δ. The vector n (2 ;  2 ; 1) is normal to (P).
Then the vectors AB(2,3,2) and u(3,2,2) are direction vectors of the plane (P).
Let's do The scalar products to check if n is normal to (P):
n • u = 3 * 2 + 2 * (– 2) + (– 2) * 1 = 6 – 4 – 2 = 0
AB • n = 2 * 2 + 3 * (– 2) + 2 * 1 = 4 – 6 + 2 = 0
n is orthogonal to u and AB, and therefore orthogonal to (P), n is a normal vector to (P).
Knowing the normal vector, the cartesian equation of (P) has the form: 2x2y+z+d=0
The plane (P) contains Delta and the point A(8,0,8) then
2*82*0+8+d=0 <=> d=24
the cartesian equation of (P) is 2x2y+z24=0
b)lets take the point M(x,y,z).
we know the coordinates of M verify the system:
x =  5 + 3 s
y = 1 + 2 s
z =  2 s
Let's calculte the distance of M to (P) with the known formula:
d(M,(P))=axM+byM+czM/sqrt(a²+b²+c²)
then
d(M,(P))= 2x2y+z24/sqrt(2²+(2)²+1)=2(5+3s)2(1+2s)2s/sqrt(9)=36/3=12
We find a real number independent of the point M. Then the distance of any point M of (D)
to (P) is independent of M. The interpretation of this result is obvious since (D) is parallel to (P).
c) To give a parametric equation of (P) with (xOy) let's resolve the system:
2x2y+z24=0
z=0
<=>
2x=2y+24
z=0
<=>
x=y+12
z=0
The system of parametic equations is:
x=t+12
y=t t=IR
z=0
3)

How far up in math did you go when you were in school? I also like math, but sadly all I could do is Calculus I.

to humber...baccalaureate +5.
This exercise was in fact not very difficult but the notation can seem weird for a beginner.
Now Im proposing a new exercise of probability that is probably a little easier:
The antitheft device of a car radio is made up of 4 numbers, each number taking a value between 0 and 9 (10 possible values)
1)a)What is the number of all the possible codes ?
b)What is the number of codes made up of 4 distinct numbers (distinct 2 by 2).
2)After a power cut, the owner of the car radio has to reintroduce its code to use it.
He knows the four number of his code are 1,9,9,5, but he has forgotten the order of these numbers.
a)How many codes can he enter with these 4 numbers?
b)If the first code entered is not good, the owner has to wait 2 minutes before entering a new one. The waiting time is doubling between two successive attempts.
How many codes can the owner test to the maximum in 24 hours?

@scarface  I envy your level of education in math. Sadly for me that wasn't possible  and not due to lack of interest or not understanding it.
The number of possible codes is 10^{4}, or 10,000. Give me an example of what you mean by distinct codes, I'm lost on that one.
Oh... I forgot. Did you post that you were now working? Do you finally have a job?

Its a good answer...perhaps Ill give a correction tonight after a beer.
As for the level of these exercises, probably you can understand the correction if you understand the notation. If they were really difficult, you couldn't.
As for the job, yes, Im now working. I worked today but I took 3 weeks leave, Im tired.

correction of the exercise:
1)
a) the number of possible codes is a 4lists of 10 numbers. so they are 10^4=10000
b)The number of different codes (0987 has four different numbers, 1223 has not) is composed
by 10numbers, then 9, then 8 then 7, every chosen number cant be taken again. So this number is 10*9*8*7=5040.
2)
a) If the number were all different, the number of different codes possible would be a permutation of 4 elements:
it would be 4!=4x3x2X1=24. (the first number can be one of the four numbers, then the second the 3 following numbers and the third the 2 following numbers...) But we have 1,9,9,5 and the number 9 is repeated. there is 2 possibility to permute 2 different numbers in a code so the numers of possibility is to divide by 2,
so the numbers of different possible codes with the numbers 1,9,9,5 is (4*3*2)/2=12
b)24hours is 24*60 minutes = 1440 minutes
if we do a chart to represent the numbers of attempt, the time to add between 2 attempts and the total waiting time, we have this:
attemptwaiting timetotal time
1 0 0
2 2 2
3 4 6
4 8 14
5 16 30
6 32 62
7 64 126
8 128 254
9 256 510
10 512 1022
11 1024 2046
after 10 attempts 1022 minutes have passed, after 11, 2046. So the answer is 10.

Perhaps Im going to come back to the first question of the exercise with vectors to give more explanations to humbert.
He seemed abashed, and I would not want him to stay traumatized.
I guess he didnt understand the terms "system of parametric equations"
In a normal plane, (Oij), we usually have reduced equation like y=ax+b. probably shadow97 knows what Im talking about, its something we can see at school.
for this kind of equation, we could use cartesian equation like yaxb=0 but its rare.
In the space,its different, we use a plane (Oijk) and a line is either defined with a system of parametric equations or a system of cartesian equations because a line is considered as an intersection of 2 planes. if you look at the vectors you see 3 numbers, we are in the space.
so lets take the first question:
we have the point A(8,0,8) and the vector AB(2;3;2), a direction vector of delta.
Then,
x = 8+2t
y = 3t t=IR
z = 8+2t
This is the system of parametric equations. This result is obvious and can be found rapidly.
now we are going to look for a system of reduced eqations using this sytem, probably more understandable for you.
So we have this system. We are going to isolate t:
t=(x8)/2
t=y/3
t=(z8)/2
therefore:
(x8)/2=y/3
y/3=(z8)/2
then
3x24=2y
2y=3z24
so we have the system of cartesian equations:
3x2y24=0
2y3z+24=0
if you dislike this notation we can write:
y=3/2x12
y=3/2z12
both equations are necessary unlike in a plane Oij where there is no z.

The last time we have seen that humbert was happy to do some mathematics. This time we are going to do some statistics.
I chose a somewhat difficult exercice, but for those who know the distribution of poisson its not.
So we have 2 variables, X and Y based on the poisson distribution with the parameters l and u: X>P(l), Y>P(u).
We are going to demonstrate that the Sum S=X+Y is based on the parameters l+u: (X+Y)>P(l+u)
Here is what I did:
(https://i.ibb.co/pzttT8c/330067WP20140104004.jpg)

Impressive, I've got an A in maths but I have close to no idea what's going on. I think I'm going to further study Maths next year. It's a good thing to know.

The last exercice in statistics is at an academic level. Probably You wont study the distribution of poisson anytime soon. The exercices in maths are easier.

This is the part of Math that I'm glad I already finished my undergrad studies. Hopefully no complicated Maths in Masteral subjects! ???
My big bro will probably be happy about this, he's really into math... I, on the other hand, choose to vow out of this situation. I'll leave you guys to it... ;)

Scarface you're a cool one. I'm recoginsing here a formulas we learnt at math statisitics at univercity. But I never can imagine that someone can play with this just for fun, like you do. :) We using this Poisson distribution in theory of digital signal processing and in calculation statistical distribution of marketing parameters.

We have seen that Daniil, like humbert was eager to do some statistics. And no matter why, it can be useful in real life, not if your only hobby is to watch TV of course.
Now we are going to have some more difficult exercises. if shadow97 wonders why he has not done that at school, its normal, perhaps he will see that if he wants to become an administrator in a statistics institute.
So here is the exercice:
X is a random variable with the following density of probability:
f(x,c)=e^((xc)) if x>c (c is an unknown real parameter)
0 otherwise
1) calculate the expected value of X
2) deduce an estimator of c based on the random vector Xn={X1,...,Xn}
is it biased or convergent?
tips: we are going to use the expected value with the law of the great numbers.

Here is a correction for the exercise.
For those who will work on that, its an opportunity because I explained the moments method used to obtain an estimator. In this case its simple though.
(https://i.ibb.co/hZrXSGd/495791WP20140126004.jpg)

@Scarface  how far up in math did you go, and how long ago did you study it? Sadly I only went as far as Calculus I, which isn't a whole lot. Bad circumstances made it impossible for me to keep studying. :(

What I did is not your business humbert. But I can assure you I was good at statistics before doing these exercices. Im doing this to eventually take some exams and move on to other jobs.
Of course, We havent finished with the last exercice yet.
Here are 2 more questions:
3) Calculate the limited distribution of the estimator ( using the theorem of the central limit).
4) Calculate the bound of CramerRao for the estimator (very difficult).
A good clip for humbert, still "roi et mage" in here:
http://www.youtube.com/watch?v=ZeBKL2hcdgk

@Scarface → I don't understand why you're upset. All I asked was how far up in math did you go in school. Of course it's possible that you studied more of it as well as statistics on your own. Was that what happened, if you don't mind telling us?

I studied mathematics...10 years ago. I just regret not to have done something useful like Ahmad. So I wont talk about the past. And dont think its easy, if I know how to resolve question 3) I have no idea how to do 4)

I studied mathematics...10 years ago. I just regret not to have done something useful like Ahmad. So I wont talk about the past. And dont think its easy, if I know how to resolve question 3) I have no idea how to do 4)
It's past midnight over there. Shouldn't you be asleep? :)
I'm like you in that I too regret that I was not able to continue studying math, physics, stats etc. It's a long story. Like you said, the past is the past, we must continue forward. And yes, hopefully Ahmad will finish medical school and graduate as a doctor.

yes it is. and Im getting up at 7 am because Im working but Im not sleepy. Perhaps because Im drinking a lot of coffee each day.

yes it is. and Im getting up at 7 am because Im working but Im not sleepy. Perhaps because Im drinking a lot of coffee each day.
Offtopic
I quite often stay up to 45AM on school days where I have to get up at 7. I really can't fall asleep at night. I go to school, and come home, sleep then do what I want for the rest of the hours.
Regarding coffee though. How much do you drink? I love coffee, strong coffee with a little milk.

Do your parents know youre talking to a guy nicknamed scarface twice as old as You during the night? You might be tired at school.

Here is the correction for the question 3. Perhaps it will be of some use for shadow97, who knows. The question 4 will be seen later, its more difficult.
(https://i.ibb.co/ByjgW9d/17676559ex.jpg)

Here Is the correction for the fourth question.
The bound FDCR Is used to calculate the lower bound of the variance of an estimator.
Previously humbert told us he was afraid his level might not be sufficient, si I guess he hasn't found the bound of FrechetDarmoisCramerRao...
This example was not good anyway as the 3rd hypothesis makes it useless to calculate it.
(https://i.ibb.co/SdydjhT/WP20140222001.jpg)

Tonight I Will suggest an easy exercice in statistics. Previously we have seen that Daniil was delighted to do some statistics, the distribution of poisson probably reminded him of a distant past where he was happy.
We know how to calculate a cumulative distribution function (cfd) when we know the probability distribution (pd) But its harder to do the contrary.
a cfd of a random variable X Is equal to 0 when x<1.
F(X)=11/(n(n+1)) for n<x<n+1 and n>=1
Question: find the pd.

Tonight I Will suggest an easy exercice in statistics. Previously we have seen that Daniil was delighted to do some statistics, the distribution of poisson probably reminded him of a distant past where he was happy.
We know how to calculate a cumulative distribution function (cfd) when we know the probability distribution (pd) But its harder to do the contrary.
a cfd of a random variable X Is equal to 0 when x<1.
F(X)=11/(n(n+1)) for n<x<n+1 and n>=1
Question: find the pd.
How do you come up with these exercises? Is there a reason to solve them or is it completely for fun?

I find them in books. You can solve them if You want, or not. It seems nobody ever tried, except perhaps humbert. I forgot to say n Is an integer, so the variable Is discrete.
for this exercice the answer Is P(X=n)=F(n+1)F(n) (You can calculate that) for n>1 and P(X=1)=1/2.

I find them in books. You can solve them if You want, or not. It seems nobody ever tried, except perhaps humbert. I forgot to say n Is an integer, so the variable Is discrete.
for this exercice the answer Is P(X=n)=F(n+1)F(n) (You can calculate that) for n>1 and P(X=1)=1/2.
I would probably try if I had the knowledge to do this. I'm not experienced enough, I still got atleast one year left of maths, probably 2 years.

Youre Lucky. Did You see me? Im an old bag and im still doing That. In fact I wish I could do something else. I told my coworkers I wanted to adopt a 20years old (I wouldnt have the time to breed a child), and he could even work and bring some money for me. They told me I was crazy.

Youre Lucky. Did You see me? Im an old bag and im still doing That. In fact I wish I could do something else. I told my coworkers I wanted to adopt a 20years old (I wouldnt have the time to breed a child), and he could even work and bring some money for me. They told me I was crazy.
Believe me, a child is an awesome responsibility. The problems you have will multiply exponentially.

Youre Lucky. Did You see me? Im an old bag and im still doing That. In fact I wish I could do something else. I told my coworkers I wanted to adopt a 20years old (I wouldnt have the time to breed a child), and he could even work and bring some money for me. They told me I was crazy.
Believe me, a child is an awesome responsibility. The problems you have will multiply exponentially.
How many do you have, if I may ask?

Actually I have no offspring. But knowing There are 7 billion people, and seeing what Is going on, Im trying to put things into perspective.
We have seen That humbert and shadow97 were abashed at the difficulty of the previous exercises. We are going to do something easier.
We are launching 2 coins and we are playing heads or tails.
what Is the proba to see 2 heads?
determine oméga (the universe)
determine card oméga (number of possibilities)

Actually I have no offspring. But knowing There are 7 billion baboons and seeing what Is going on, Im trying to put things into perspective.
We have seen That humbert and shadow97 were abashed at the difficulty of the previous exercices. We are going to do something easier.
We are launching 2 coins and we are playing heads or tails.
what Is the proba to see 2 heads?
determine oméga (the universe)
determine card oméga (number of possibilities)
 25%? 1/4th
O 50% X 1/2 O = heads, X = tails
/ \ / \
25% O O 25% 25% O X 25 % 1/4
 8 possibilities
I don't really understand what you mean by omega, sorry ^^ I hope this is somewhat correct, if so this is what we did last year.
Currently doing my maths homework:
I can write out my excersices. My homework is due tomorrow, soo yup.
1; You have the equation x  18 = 22.
A: Which number should you add to both the sides to get x free? (I will do my best to translate)
B: Which solution does the equation have?
2; You have the equation x + 13 = 47.
a: Which number should you subtract from both the sides to get x free?
b: Which solution does the equation have?
3; You have the equation 4 * x = 22.
a: With which number should you divide both the sides to get x free?
b: Which solution does the equation have?
4; You have the equation x/7 = 15.
a: With which number should you multiply both the sides to get x free?
b: Which solution does the equation have?
^Not hard in my opinion, that is basically what you need to get the grade E.
9; Which of the following steps should be executed to solve the equation and in which order?
{A: Add 5 to both sides.}
{B: Subtract 5 from both the sides.}
{C: Multiply both the sides with 5.}
{D: Divide both the sides with 5.}
a: 5x + 5 = 30
b: 5x  5 = 3
c: x/5  5 = 20
d: x + 5/ 5 = 3
I have no issues with these assignments, I think that they are too easy for me.
I just find it annoying that i have already solved all of this, but I need to turn it in now, 3 weeks later because I've worked too far ahead in the maths book. And, ofcourse I have trashed my paper with the solutions 8)

Omega Is a greek letter usually standing for the universe, or fundamental ensemble.
here Is the correction:
if A Is the event "2 heads", P(A)=1/4.
omega={H,T}^2
card omega=3 (dont know how You found 8 )
The exercices You have are easy indeed.

For Omega you could have typed Alt 234 Ω or Alt +3A9 Ω  Ω Ω Ω and of course "squared" is alt 253 ² :) :)

Omega Is a greek letter usually standing for the universe, or fundamental ensemble.
here Is the correction:
if A Is the event "2 heads", P(A)=1/4.
omega={H,T}^2
card omega=3 (dont know how You found 8 )
The exercices You have are easy indeed.
I got no idea what I am doing ;D

For Omega you could have typed Alt 234 Ω or Alt +3A9 Ω  Ω Ω Ω and of course "squared" is alt 253 ²
Under Windows phone, no alt nor ², sorry.
We can note That shadow97 found successfully the first question, though hes probably not learning that at school.

Under Windows phone, no alt nor ², sorry.
I've never tested Windows Phone, but if there's no way to insert special characters that's pretty lame. Many Android keyboard programs have a good number of special characters, plus there are Unicode program that allow you to copy and paste them, just like Character Map on Windows.

A long time ago I remember that humbert told me that he could learn the French language in 6 months. And he's probably right since a lot of words are close in English and French. It's partly due to the fact that England has had French kings in the past. I also learnt Spanish and despite that both language have Latin roots, it was quite hard to learn it. For example compare carotte in French, carrot in English and...zanahoria in Spanish.
And some people here should be proud to master English as well as they do. Others still have to improve it though.
Anyway I wanted to show humbert, who is very gifted, that speaking French is not enough. It's an incredibly difficult language if you want to write it  without fault.
So I'm going to give humbert a few easy examples to show him why it's difficult.
For him it will be an easy class.
Let's begin.
Les participes passés utilisés avec l’auxiliaire être s'accordent, alors qu'ils ne s'accordent pas avec l'auxiliaire avoir.
ainsi compare:
Les babouins ont mangé (pas d'accord)
Les babouins sont tombés (accord)
Note que contrairement à l'anglais, le pluriel n'a pas d'impact sur la prononciation.
par exemple, babouin  babouins se prononcent de la même façon. Ce n'est pas le cas en anglais: baboon  baboons.
Ce n'est donc pas obligatoire de bien connaître la grammaire pour avoir une prononciation correcte des mots, même si certains pluriels sont liés, entre un article et le nom qui le suit notamment. Par exemple pour "les éléphants", la prononciation correcte implique de dire un "z" entre les 2 mots, puisque "éléphants" commence par une voyelle. Mais il y a des exceptions. Par exemple on dira "des arbres" avec "z" mais pour "des haricots" il n'y a pas de liaison, le h est dit "muet". Beaucoup de personnes ignorent cette règle et le prononcent cependant.
maintenant plus difficile, avec l'accord des verbes pronominaux.
imagine que le sujet "Je" est une femme.
Je me suis blessée (accord, le verbe s'accorde avec le pronom me, qui représente la femme, placé avant le verbe)
Je me suis blessé la main (pas d'accord: la femme s'est blessé quoi? la main, qui est objet direct et placée après le verbe).
C'est à la main que je me suis blessée. (accord: le participe passé blessée est accordé avec la main, nom féminin et placé avant le verbe.
Bien sur il y a des exceptions et certains verbes ne suivent pas la règle.
compare:
elles se sont blessées (accord car le pronom "s' " est objet direct est placée avant le verbe)
elles se sont succédé (pas d'accord car le pronom "se" est un objet indirect: elles ont succédé à ellemême  forme pas très correcte)
Note que normalement il n'y a pas d'accord avec l'auxiliaire avoir. Mais si le complément est direct et qu'il est placé devant, le participe passé s'accorde...
par exemple:
Ils ont acheté des teeshirts (pas d'accord).
Ce sont les teeshirts qu'ils ont achetés (accord)
Plus difficile: quand le verbe pronominal est suivi par un infinitif.
Compare:
elle s'est laissée mourir. (accord: elle est le sujet)
elle s'est laissé rejoindre (pas d'accord: elle n'est pas le sujet réel, quelqu'un l'a rejointe)
la règle est la même avec l'auxiliaire avoir:
c'est la femme que j'ai entendue chanter (entendu est accordé car c'est la femme qui chante)
ce sont les fruits que j'ai vu cueillir ('vu' n'est pas accordé car les fruits ne font pas l'action)
ce sont les fruits que j'ai vus tomber ('vu' est accordé car ce sont les fruits qui tombent)
Note que sans verbe à l'infinitif à la fin de la phrase, c'est la règle décrite avant qui fonctionne....
par exemple:
ce sont les fruits que j'ai vus (il y a accord obligatoire du participe passé 'vu')

A long time ago I remember that humbert told me that he could learn the French language in 6 months. And he's probably right since a lot of words are close in English and French. It's partly due to the fact that England has had French kings in the past. I also learnt Spanish and despite that both language have latin roots, it was quite hard to learn it. For example compare carotte in French, carrot in English and...zanahoria in Spanish.
And some people here should be proud to master English as well as they do. Others still have to improve it though.
Anyway I wanted to show humbert, who is very gifted, that speaking French is not enough. It's an incredibly difficult language if you want to write it  without fault.
So I'm going to give humbert a few easy examples to show him why it's difficult.
For him it will be an easy class.
Let's begin.
Les participes passés utilisés avec l'auxilaire être s'accordent, alors qu'ils ne s'accordent pas avec l'auxiliaire avoir.
ainsi compare:
Les babouins ont mangé (pas d'accord)
Les babouins sont tombés (accord)
maintent plus difficile, avec l'accord des verbes pronominaux.
imagine que le sujet "Je" est une femme.
Je me suis blessée (accord, le verbe s'accorde avec le pronom me, qui représente la femme, placé avant le verbe)
Je me suis blessé la main (pas d'accord: la femme s'est blessé quoi? la main, qui est objet direct et placée après le verbe).
C'est à la main que je me suis blessée. (accord: le participe passé blessée est accordé avec la main, nom féminin et placé avant le verbe.
Bien sur il y a des exceptions et certains verbes ne suivent pas la règle.
compare:
elles s'est blessée (accord car le pronom "s' " est objet direct est placée avant le verbe)
elles se sont succédé (pas d'accord car le pronom "se" est un objet indirect: elles ont succédé à ellemême  forme pas très correcte)
Note que normalement il n'y a pas d'accord avec l'auxiliaire avoir. Mais si le complément est direct et qu'il est placé devant, le participe passé s'accorde...
par exemple:
Ils ont acheté des teeshirts (pas d'accord).
Ce sont les teeshirts qu'ils ont achetés (accord)
Plus difficile: quand le verbe pronominal est suivi par un infinitif.
Compare:
elle s'est laissée mourir. (accord: elle est le sujet)
elle s'est laissé rejoindre (pas d'accord: elle n'est pas le sujet réel, quelqu'un l'a rejoint)
la règle est la même avec l'auxiliaire avoir:
c'est la femme que j'ai entendue chanter (entendu est accordé car c'est la femme qui chante)
ce sont les fruits que j'ai vu cueillir (vu n'est pas accordé car les fruits ne font pas l'action)
ce sont les fruits que j'ai vus tomber (vu est accordé car ce sont les fruits qui tombent)
this is a good tutorial. for a while i'm just reading and when i'm sure will joined
so i wanted to know also humbert. merci beaucoup

When I was saying that some should do efforts to speak English, I was also thinking about you iih. Of course you can try to learn French, if you have some questions (des questions intéressantes, c'est évident), you can ask me, but first you should try to do efforts to improve what you write here. Read again some of your posts and you will notice that some are hardly understandable, perhaps even for you ("when i'm sure will joined"=> is it meaning anything?).

There is nothing difficult about Spanish. You read and write it exactly the same way you pronounce it. Vowels have only one sound  their own name  in every and all situations. Grammar rules aren't too complicated. It's almost like Esperanto. By comparison English and French are nightmares. In French you have many [for example] 8 letter words, and 5 of them are silent. But at least in French, just like in Spanish, you have accented vowels and extended characters to help. There's nothing like this in English. You have words such as "appropriate" which, if pronounced "apropriāt" means one thing, but pronounce it "appropriit" and it's another definition entirely. There is NO accent or extended character to tell you which is which. And as in French, you have plenty of silent letters.

Tonight I'm going to give you a French lesson, and we have seen that the last time I taught a course, iih was very happy to attend the class. Maybe teachers are scarce in his country and internet is a good way to learn some things. At least, personally I learnt many things thanks to my computer. I wasn’t bilingual 10 years ago, but reading some newspapers and watching some movies in English have been really helpful. I stay lucid, it won't be enough to read the newspaper le Monde, but you may learn a few things. What's more, the French language is only spoken in France,Switzerland,Canada,North Africa, and less and less in Vietnam. So it is unlikely that Vasudev or iih will have to speak French in a discussion one day.
(http://copinette.c.o.pic.centerblog.net/tfkmv7qj.gif)
Maybe those with gaps in English are going to learn a few words, even though we are going to see some common words. Tonight's lesson will focus on false friends.
A false friend is a word that looks similar but whose meaning is different.
For example, let’s take the word Important: in French it means both significant and important:
une personne important/an important person. une somme important/a significant sum (an important sum would not really make sense).
So on the left I put the French words, and on the right the English translation.
Important Important/significant
Librairie Book shop (bibliothèque=library)
Actuellement Currently (en fait=actually)
Éventuellement Possibly
Fastidieux boring (Méticuleux= fastidious/meticulous)
Habileté skill (and not ability)
Réaliser to achieve/to realize (Achever= to complete)
Hasard chance (Danger = hazard/danger)
Heurter to hit (Blesser= to hurt)
Prétendre to claim (Faire semblant=to pretend)
Raisonnable sensible (sensible=sensitive)
Approvisionner to supply (supplier=to beg)
Impoli rude (rude=hard/rough)
Nom de famille surname (surnom=nickname)
Expression phrase (phrase=sentence)
Monnaie change (changer=to alter/to change)
Grandiose grand (grand=tall)
Grue crane (crane=skull)
Tissu fabric (fabrique=factory)
Exiger to demand (demander=to ask)

Some persons must be getting a bit bored on the forum...I was doing some accounting exercises and suddenly I was thinking “maybe shadow97 or usmangujjar are eager to do these exercises too”.
So, here are a few question about amortization.
Nomaher.com company bought a new patent for $25,000. Because of government regulation the patent is now worth $35000. The process is only expected to be used for 5 years. The annual amortization expense associated with this patent is
A $2,000
b. $5,000
c. $7,000
d. $10,000
The estimate of the value of an asset at the time of disposal is the
a. acquisition cost.
b. residual value.
c. depreciable cost.
d. cost of living.
Depreciation expense is recorded in order to
a. ensure that the firm has adequate cash to replace the asset.
b. present assets at fair market value.
c. be able to sell the asset easily when the firm does not need it any more.
d. allocate the cost of noncurrent, nonmonetary assets to periods benefited.
You can reply and I'll tell you if your answers are right or wrong...
And here is an exercise about the declining balance method, which is a widely used form of accelerated depreciation.
You have to do an amortization table. Well you’ll find a model below, and if you want to reply you just have to fill it. Note that for the US rule, the last depreciation must take into account the residual value. It’s not the case in the French accounting system where only the linear method is taking into account the residual value.
Cost of asset = $40,000
Estimated residual value = $4,000
Estimated useful life of asset = 5 years
Depreciation rate = (1/useful life) x 2=40%
Year Cost Depreciation Rate Depreciation Accumulated Depreciation Book Value
1 40000 40%
2 40000
3 40000
4 40000
5 40000

Today, I'm going to suggest a philosophy test.
Test your knowledge with these 5 philosophy questions. The answers are below.
1 "Beauty is the symbol of good" according to:
 Hume
 Plato
 Kant
2 What idea was John Stuart Mill behind?
 Better to be a dissatisfied man than a dissatisfied pig "
 Stop and think about the meaning of your actions ”
 Man is neither angel nor beast, and misfortune dictates that whoever wants to be an angel plays the beast "
3 According to Hobbes, the three main causes of quarrels are: rivalry, mistrust and…
Pride
Jealousy
 Ignorance
4 According to Sartre, "man is condemned to ...":
 Die
 Be free
 Live
5 According to Plato, to know the real it is necessary:
 To isolate yourself in a cave
 To detach from the senses
 To look at the stars
The Thinker by Auguste Rodin, in the museum Rodin, near the Invalides, in Paris.
(https://i.ibb.co/h88Q6Rp/thinkeraugusterodinmuseumparis1904.jpg)
1 "Beauty is the symbol of good" according to Kant, in the Critique of the faculty of judging. As Lamouche, Rosset and Cerqueira write, “if the beautiful interests us, says Kant, it is because it presents an analogy with the good: just as the concern for the good relegates our interest to the background, the beautiful brings us into a disinterested pleasure distinct from the pleasant ”.
2 For John Stuart Mill “Better to be a dissatisfied man than a dissatisfied pig; it is better to be a satisfied Socrates than a satisfied fool ".
3 According to Hobbes, the three main causes of quarrels are rivalry, mistrust and… pride. "It becomes clear by this that as long as men live without a common power that holds them all at bay, they are in this condition which is called war, and this war is war of each against each" (Leviathan).
4 According to Sartre, "Man is doomed to be free. Sartre writes this in Being and Nothingness. Man, who is absolutely free, "is always in a position of choice, and therefore of responsibility, which plunges him into anxiety and forbids him lightness", explain Lamouche, Rosset and Cerqueira.
5 According to Plato, to know the real one must detach oneself from the senses. "The soul of the true philosopher is kept away from pleasures, passions, sorrows, fears, as far as it is possible" (Phaedo).