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Total traded size

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The number $N$ of trades in an hour is Poisson with mean $5$ . Each trade has size $1$ , $2$ or $3$ with equal probability, independently of the other trades and of $N$ . What is the variance of the total size traded in the hour?

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Hint

Use the variance of a random sum: it depends on the mean and variance of N and on the trade size moments.

Solution

One trade size has mean $2$ and variance $\frac{1 + 4 + 9}{3} - 4 = \frac{2}{3}$ .
By the random-sum formula, $Var (S) = E [N] Var (Y) + Var (N) E [Y]^{2}$ .
This is $5 \times \frac{2}{3} + 5 \times 4 = \frac{10}{3} + 20$ , which equals $\frac{70}{3}$ .

Answer: $\frac{70}{3}$

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