Calculate Expectation and variance of variable that is average of $n$ random variables

Question

Suppose $Y_1, Y_2, \ldots , Y_n$ are all independent random variables with mean $a$ and variance $b$. Define Tn to be their average, i.e.

$$T_n = \frac{Y_1 + Y_2 + \cdots + Y_n} n$$

Calculate $\operatorname{E}(T_n)$ and $\operatorname{Var}(T_n)$, in terms of $a$ and $b$.

Find the smallest integer value of $n$ such that $T_n$ is “close enough” to $\operatorname{E}(T_n)$, in the sense that

$$|T_n − \operatorname{E}(T_n)| ≥ \sqrt2 \, b$$ occurs with probability at most $4\%.$

Answer 1

For the first part we have

$$E(T_n)=\frac{E(Y_1)+\dots+ E(Y_n)}{n}=\frac{an}{n}=a$$

For the second part we have:

$Var(T_n)=E(T_n-E(T_n))^2=E((\frac{Y_1+Y_2+\dots+Y_n}{n})^2-2a^2+a^2)=\frac{n-1}{n}E(Y_1Y_2)+\frac{1}{n}E(Y_1)^2 -a^2=\frac{n-1}{n}a^2+\frac{1}{n}(b+a^2)-a=\frac{b}{n}$