Total daily signups from two channels

Question

A website gets new signups from ads as a Poisson stream averaging 3 per day and, independently, from organic search as a Poisson stream averaging 5 per day. Let $S$ be the total number of signups in a day.

What distribution does $S$ follow, and what are its mean and variance?

Accepted Answer

Independent Poisson streams superpose: adding them gives another Poisson whose rate is the sum of rates. And a Poisson always has mean equal to variance. So S \sim 	ext{Poisson}(3 + 5) = 	ext{Poisson}(8), giving \mathbb{E}[S] = 8, \qquad \operatorname{Var}(S) = 8. Standard deviation \sqrt{8} \approx 2.83. Takeaway. Pool independent Poisson sources by adding rates (here to 8), and mean equals variance for the pooled count.

Total daily signups from two channels

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