Question

Let Xi,.,Xn be independent random variables with common probability density f(x) = ה sin(x) , x E [0, π] (a) Assuming EX,] = 2, calculate Var(X). (b) Assuming Var(Xs + + X,) = Var(X) + Var(Xn) and if a, b є R that Var(ax, + b) = a2Var(X), calculate the mean and the variance of Zn, defined [1 as follows: Var(X1+...+ Xn)

Answer 1

Since the PDF is so

$E(X)=\int_{0}^{\pi}xf(x)dx\\~~~~~~~~~~~~~={1\over 2}\int_{0}^{\pi} x\sin xdx\\ ~~~~~~~~~~~~~={1\over 2}\left ( x(-\cos x ) -(-\sin x)\right )_{0}^{\pi}\\ ~~~~~~~~~~~~~={\pi\over 2}$

and

$E(X^2)=\int_{0}^{\pi}x^2f(x)dx\\~~~ ~~~~~~~~~~={1\over 2}\int_{0}^{\pi} x^2\sin xdx\\ ~~~~~~~~~~~~~={1\over 2}\left ( x^2(-\cos x ) -2x(-\sin x)-2\cos x\right )_{0}^{\pi}\\ ~~~~~~~~~~~~~={1\over 2}(\pi^2+4)$

Hence $Var(X)=E(X^2)-[E(X)]^2={1\over 2}(\pi^2+4)-\left ( \pi\over 2 \right )^2={\pi^2\over 4}+2$

a) Hence   $E(X_1)={\pi\over 2}$ ,   $Var(X_1)={\pi^2\over 4}+2$

b) Since $E(X)={\pi\over 2}$ and $Var(X)={\pi^2\over 4}+2$ so

$E(X_i)={\pi\over 2}~,~Var(X_i)={\pi^2\over 4}+2$ for all $i=1,2,...,n$

Hence $E(X_1+\cdots+X_n)=n{\pi\over 2}$ and $Var(X_1+\cdots+X_n)=n\left ({\pi^2\over 4}+2 \right )$

Now using properties of expectation and variance we have

$E(Z_n)=E\left ({X_1+\cdots+X_n-E[X_1+\cdots+X_n]\over \sqrt{Var(X_1+\cdots+X_n)}} \right )\\ ~~~~~~~~~~~~~~={E(X_1+\cdots+X_n)-E[X_1+\cdots+X_n]\over \sqrt{Var(X_1+\cdots+X_n)}}\\ ~~~~~~~~~~~~~~={0\over \sqrt{Var(X_1+\cdots+X_n)} }\\ ~~~~~~~~~~~~~~=0$

and

$Var(Z_n)=Var\left ({X_1+\cdots+X_n-E[X_1+\cdots+X_n]\over \sqrt{Var(X_1+\cdots+X_n)}} \right )\\ ~~~~~~~~~~~~~~={Var(X_1+\cdots+X_n)\over {Var(X_1+\cdots+X_n)}}\\ ~~~~~~~~~~~~~~=1$