Question

Given the following discrete uniform probability distribution, find the expected value and standard deviation of the random variable. Round your final answer to three decimal places, if necessary.

Probability Distribution
x 0 1 2 3 4 5 6 7 8 9 10
P(X=x) 1/11 1/11 1/11 1/11 1/11 1/11 1/11 1/11 1/11 1/11 1/11

Answer 1

The expected value is

$E[X]=\displaystyle\sum_xx\,P(X=x)=\frac1{11}\sum_{x=0}^{10}x=\dfrac{0+1+\cdots+9+10}{11}=\dfrac{55}{11}=\boxed{5}$

The standard deviation is the square root of the variance, which is

$V[X]=E[(X-E[X])^2]=E[X^2]-E[X]^2$

where

$E[X^2]=\displaystyle\sum_xx^2\,P(X=x)=\frac1{11}\sum_{x=0}^{10}x^2=\dfrac{0^2+1^2+\cdots+9^2+10^2}{11}=\dfrac{385}{11}=35$

so that

$V[X]=35-5^2=10$

making the standard deviation

$\sqrt{V[X]}=\sqrt{10}\approx\boxed{3.16}$