Question

3. Suppose that X has a discrete uniform distribution on the integers 0 through 9.

Determine the mean, variance, and standard deviation of the random variable Y = 5X

and compare to the corresponding results for X.

Answer 1

We have

$EX=\sum{x_ip_i}=0\cdot 0.1+1\cdot 0.1+...+9\cdot 0.1=4.5\\EX^2=\sum{{x_i}^2p_i}=0^2\cdot 0.1+1^2\cdot 0.1+...+9^2\cdot 0.1=28.5\\DX=EX^2-\left( EX \right) ^2=28.5-4.5^2=8.25\\\sigma X=\sqrt{DX}=\sqrt{8.25}=2.87228\\EY=E\left( 5X \right) =5EX=5\cdot 4.5=22.5\\DY=D\left( 5X \right) =25DX=25\cdot 8.25=206.25\\\sigma Y=\sqrt{DY}=\sqrt{206.25}=14.3614$