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Gap between $2$ draws

Expected valueMedium

$2$ integers are drawn independently and uniformly from $1$ to $5$ , and the payout is the absolute difference between them. What is the variance of the payout?

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Hint

Count how many of the equally likely pairs give each difference.

Solution

Of the $25$ equally likely pairs, the differences $0$ , $1$ , $2$ , $3$ and $4$ occur $5$ , $8$ , $6$ , $4$ and $2$ times.
So $E [X] = \frac{40}{25} = \frac{8}{5}$ and $E [X^{2}] = \frac{100}{25} = 4$ .
The variance is $4 - (\frac{8}{5})^{2} = 4 - \frac{64}{25} = \frac{36}{25}$ .

Answer: $\frac{36}{25}$

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