Mean and variance of hit RFQs (binomial)

Question

A market maker responds to 50 independent requests-for-quote (RFQs) in a session. Each RFQ is independently "hit" (the client trades) with probability $p = 0.2$ . Let $H$ be the number of RFQs that get hit.

What are the mean and variance of $H$ ?

Accepted Answer

The count of independent yes/no events, each with the same probability, is a binomial. Here H \sim 	ext{Binomial}(n=50,\ p=0.2). Two facts you can just remember for a binomial: - Mean (expected count): \mathbb{E}[H] = np. Intuition: each of the n tries adds p on average, so n tries add np. - Variance: \operatorname{Var}(H) = np(1-p). Each try contributes a little wiggle of size p(1-p), and independent wiggles add up. Plug in: \mathbb{E}[H] = 50 	imes 0.2 = 10, \qquad \operatorname{Var}(H) = 50 	imes 0.2 	imes 0.8 = 8. The standard deviation is \sqrt{8} \approx 2.83, so a typical session l…

Mean and variance of hit RFQs (binomial)

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