Question

A fair spinner split evenly into 4 spaces has the values 2,5,6 and 10. find the variance

Answer 1

Let
`X` denote the value the spinner lands on. Then
`X` has PMF

`P(X=x)=\begin{cases}\frac14&\text{for }x\in\{2,5,6,10}\\\\0&\text{otherwise}\end{cases}`

Then the average value the spinner lands on is

`E[X]=\displaystyle\sum_xx\,P(X=x)=\frac{2+5+6+10}4=\frac{23}4`

and the variance is

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

We have

`E[X^2]=\displaystyle\sum_xx^2\,P(X=x)=\frac{2^2+5^2+6^2+10^2}4=\frac{165}4`

so that the variance is

`V[X]=\frac{165}4-\frac{23}4=\frac{71}2`