Q.4.8 - A First Course in Probability

Question

Let $X$ be a random variable having expected value $\mu$ and variance $\sigma 2$. Find the expected value and variance of $Y=\frac{X-\mu }{\sigma }$.

Step-by-Step Solution

Answer

Mean is $0$, and variance is $1$.

Step 1: Given information

Let $X$ be a random variable having expected value $\mu$ and variance $\sigma 2$. 

Step 2: Calculation

Utilizing the linearity of expectation, we have that

$E\left(\frac{Y-\mu }{\sigma }\right)=\frac{1}{\sigma }\left(E\right(Y)-E(\mu \left)\right)=\frac{1}{\sigma }(\mu -\mu )=0$

And using the effects of the Variance (adding a constant accomplishes not change the variance and multiplying by a constant increase the variance by the square) we have that

$Var\left(\frac{Y-\mu }{\sigma }\right)=\frac{1}{{\sigma }^2}Var\left(Y\right)=\frac{1}{{\sigma }^2}\times {\sigma }^2=1$

Step 3: Final answer

The expected value and variance are $0,1$.