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

$P(X=x)=\begin{cases}\dfrac14&\text{for }x\in\{2,5,6,10}\\\\0&\text{otherwise}\end{cases}$

Then the average value the spinner lands on is

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

and the variance is

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

We have

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

so that the variance is

$V[X]=\dfrac{165}4-\dfrac{23}4=\dfrac{71}2$