We've been given the following problem:
Let $X_1,X_2$, . . . be independent, identically distributed (i.i.d) random variables with $E[X] = 2$ and $var(X)=9$, and let $Y_i = \frac{X_i}{2^i}$. We also define Tn and An to be the sum and the sample mean, respectively, of the random variables Y1, . . . , Yn. In other words, $T_n = \sum_{i = 1}^{n} Y_i$ and $A_n = \sum_{i = 1}^{n} \frac{1}{n}Y_i = \frac{1}{n}T_n$.
The first question is to "Evaluate the mean and variance of Yn, Tn, and An."
I'm a little unsure how to do this without having real numbers. The only thing I can think of is to substitute in the mean and variance of X.
This would give me $E[Y_n] = \frac{2}{2^n}$, $var(Y_n) = \frac{9}{2^n}$ and $E[T_n] = 2(1-1/2^n)$
But this seems like its too simple? and I'm not sure how I'd go about finding the variance in the sums? Would anyone mind possibly pointing me int he right direction?
Thank you!
Your calculation of the mean of $Y_n$ is fine.
The variance is not quite right. Recall that the variance of $cW$ is $c^2$ times the variance of $W$.
For calculating the variance of the sums, use the fact that a sum of independent random variables has variance equal to the sum of the individual variances.