Stochastic Calculus: Conditional Expectation

[WMDhamnekar]

Homework Statement

Suppose $X_1,X_2 \dots$ are independent random variables with $\mathbb{P}[X_j= 1] =1- \mathbb{P}[X_j=-1]=\frac13$ Let $S_n = X_1 +\dots +X_n$ Find $\mathbb{E}[S_n], \mathbb{E}[S^2_n], \mathbb{E}[S^3_n]$

Relevant Equations

Not applicable

Are my following answers correct?

 

[mjc123]

Have you tried the simple expedient of substituting some values of n and seeing if you get the right answer? Try n=1 to start.
You are making some careless mistakes
e.g. the expression for E(S²) should be n(1) + n(n-1)(1/9)
Your statement about the product of 3 random variables makes no sense; they are certainly not equal to zero. Each of the 3 random variables may independently take either of the values 1 or -1.

 

[WWGD]

Homework Statement:: Suppose $X_1,X_2 \dots$ are independent random variables with $\mathbb{P}[X_j= 1] =1- \mathbb{P}[X_j=-1]=\frac13$ Let $S_n = X_1 +\dots +X_n$ Find $\mathbb{E}[S_n], \mathbb{E}[S^2_n], \mathbb{E}[S^3_n]$
Relevant Equations:: Not applicable

Are my following answers correct?View attachment 323141
View attachment 323142
View attachment 323157

I agree with mjc -- hey, it rhymes!

You can't treat a multinomial $( X_1+X_2+....+X_n)^2$ as a standard binomial $( X_1+X_2)^2$. Look up multinomial coefficients.

 

[WMDhamnekar]

Have you tried the simple expedient of substituting some values of n and seeing if you get the right answer? Try n=1 to start.
You are making some careless mistakes
e.g. the expression for E(S²) should be n(1) + n(n-1)(1/9)
Your statement about the product of 3 random variables makes no sense; they are certainly not equal to zero. Each of the 3 random variables may independently take either of the values 1 or -1.

So, taking into consideration your this reply, I correct my amswers as follows:
$\mathbb{E}[S_n]=-\displaystyle\frac{n}{3}, \mathbb{E}[S^2_n]= n +\displaystyle\frac{n(n-1)}{9}, \mathbb{E}[S^3_n ] = -\displaystyle\frac{n}{3}-\frac{n(n-1)}{2} -\displaystyle\frac{n(n-1)}{9}$

Now, are these above answers correct?

 

[WMDhamnekar]

Correct answers are

 

[mjc123]

No, they are not. The answer for E(S_n²) is wrong for n = 2, and that for E(S_n³) is wrong for n = 3.

The answer you gave in post #4 for E(S_n²) is right, but the answer for E(S_n³) is not. You didn't show your working, but I suspect the mistake may lie in enumerating the terms of different kinds.
There are n terms of the form X_i³, each of which has expectation -1/3.
There are 3n(n-1) terms of the form X_i²X_j, each of which has expectation -1/3*1.
There are n(n-1)(n-2) terms of the form X_iX_jX_k, each of which has expectation -1/27.
(Check that the total number of terms is n³.)

 

[WWGD]

Sorry to insist on this, but I suggest you check multinomial coefficients , to determine how to expand multinomials $( x_1+x_2+...+x_k)^n$. I suspect the errors may be partially due to this.