Regarding indicators in Statistics.

[peripatein]

Hi,

Homework Statement

n different balls are distributed independently between m boxes with unlimited capacity each. I am asked to find the expectation and variance of the number of empty boxes.

Homework Equations

The Attempt at a Solution

The probability of i-th box being empty at the end is (1-1/m)ⁿ. Ergo, E[X_i] = P(X_i) = (1-1/m)ⁿ. Hence, E[X] = m(1-1/m)ⁿ.
As for the Variance, I used Var(X_i) = E[X_i](1-E[X_i])=(1-1/m)ⁿ(1-(1-1/m)ⁿ). Therefore, Var(X) = m*Var(X_i).
Is that correct?

 

[peripatein]

I am wondering why no one has yet replied. Is my formulation inappropriate/incomprehensible?

 

[Office_Shredder]

If the X_i were independent, then your variance would be correct. They aren't though, so you will have terms like E(X_i X_j) that you will have to deal with (luckily it's not too hard to calculate these) when you expand E(X²)

 

[peripatein]

So should it be:
m(1-1/m)ⁿ(1-(1-1/m)ⁿ) + 2m(1-2/m)ⁿ?
(I used the expression for the Covariance)

 

[Ray Vickson]

I am wondering why no one has yet replied. Is my formulation inappropriate/incomprehensible?

Not incomprehensible, just sloppy and incomplete. What is the meaning of $X_i$? If you mean that $X_i = 1$ if box i is empty and $X_i = 0$ if box i is not empty, then you should say so. Also, the notation $P(X_i)$ is meaningless; if you mean $P(X_i = 1)$ you should write that.

 

[peripatein]

I am sorry, I was using my mobile to post that one. I did mean everything you thought I might have meant.
I'd very much appreciate it if you could comment on my attempt at solution now.
Actually, both this one and the Statistics problem I posted earlier.

 

[peripatein]

I'd truly appreciate some feedback on my recent attempt at solution, namely:
The probability of the i-th box being empty at the end is (1-1/m)n. Ergo, E[Xi] = P(Xi) = (1-1/m)n. Hence, E[X] = m(1-1/m)n.
As for the Variance, I used the following:
Var(X) = m*V(Xi) + 2*SIGMA(where i<j)*Cov(Xi,Xj) = m(1-1/m)ⁿ(1-(1-1/m)ⁿ) + mC2*(1-2/m)ⁿ
I am not sure this is correct but would certainly appreciate any comments.