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 $4\%.$

  • $\begingroup$ did you manage to find the expectation? $\endgroup$ – Jorge Fernández Hidalgo Nov 3 '17 at 16:58
  • $\begingroup$ no, I dont know how to proceed with this. $\endgroup$ – msk3002 Nov 3 '17 at 17:00
  • $\begingroup$ use the linearity of the expectation $\endgroup$ – Jorge Fernández Hidalgo Nov 3 '17 at 17:01
  • $\begingroup$ Questions posted here should not be phrased in language suitable for assigning homework. $\endgroup$ – Michael Hardy Nov 3 '17 at 17:08

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:

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.