# Poisson distribution where each event can lead to different outcomes

I'm trying to tackle the following problem.

Suppose that customers arriving at a bank follow a Poisson distribution with rate λ=5 every 15 minutes. For every customer, the probability that he performs one transaction is 0.4, two transactions 0.3, three transactions 0.2 and four transactions 0.1. Find the expected value and the variance of the number of transactions that will be performed in a 3 hour interval.

My starting point was to calculate that, per customer the expected value of transactions is $0.4\cdot1+0.3\cdot2+0.2\cdot3+0.1\cdot4=2$

Additionally, if X is the number of customers in a 3 hour interval, $X \sim \operatorname{Pois} \left({5\cdot12}\right)$. So the expected value is 60 customers in 3 hours, in other words 120 transactions in 3 hours. However, I wasn't able to figure out how to calculate the variance of the number of transactions. Does the number of transactions in 3 hours also follow the Poisson distribution or am I missing something?

• Yes it does follow the poisson distribution as well, in which case the variance is the same as the expectation. Jun 5, 2015 at 14:32

Let us define $X$ as the random number of customers arriving in a 3 hour interval, so $X \sim \operatorname{Poisson}(\lambda = 60)$ as you noted. Now, for each $i = 1, 2, \ldots, X$, let $T_i$ be the number of transactions that the $i^{\rm th}$ customer to arrive makes, so that $$\Pr[T_i = k] = 0.1 (5 - k), \quad k = 1, 2, 3, 4.$$ We can now calculate $$\operatorname{E}[T_i] = \sum_{k=1}^4 \frac{k(5-k)}{10} = 2, \quad \operatorname{E}[T_i^2] = \sum_{k=1}^4 \frac{k^2(5-k)}{10} = 5,$$ so that $\operatorname{Var}[T_i] = 5-2^2 = 1$. Now we let $$S = \sum_{i=1}^X T_i$$ be the total number of transactions observed in the 3 hour interval. We wish to compute $\operatorname{E}[S]$ and $\operatorname{Var}[S]$. To this end, observe that

\begin{align*} \operatorname{E}[S] &= \operatorname{E}_X[\operatorname{E}[S \mid X]] \\[8pt] &= \operatorname{E}_X\left[\sum_{i=1}^X \operatorname{E}[T_i]\right] \\[8pt] &= \operatorname{E}[X \operatorname{E}[T_i]] \\[8pt] &= \operatorname{E}[X]\operatorname{E}[T_i] \\[8pt] &= 2\lambda = 120. \end{align*}

This is the law of total expectation. Next, we use the law of total variance:

\begin{align*} \operatorname{Var}[S] &= \operatorname{E}[\operatorname{Var}[S \mid X]] + \operatorname{Var}[\operatorname{E}[S \mid X]] \\[8pt] &\overset{\text{ind}}{=} \operatorname{E}[X \operatorname{Var}[T_i]] + \operatorname{Var}[X \operatorname{E}[T_i]] \\[8pt] &= \operatorname{E}[X]\operatorname{Var}[T_i] + \operatorname{E}[T_i]^2 \operatorname{Var}[X]. \end{align*}

Now it is a simple matter to recall that $\operatorname{E}[X] = \operatorname{Var}[X] = \lambda$ for a Poisson distribution, and substitute the expectation and variance for $T_i$ that we computed above. Note that this last formula $$\operatorname{Var}[S] = \operatorname{E}[X]\operatorname{Var}[T_i] + \operatorname{E}[T_i]^2 \operatorname{Var}[X]$$ applies to any random variable $S = T_1 + T_2 + \cdots + T_X$ where all the $T_i$s are iid, and $X$ is a nonnegative integer-valued random variable. This formula is frequently found in actuarial contexts.

• Thank you for the very detailed answer. Jun 5, 2015 at 19:29
• how is 5 - 2^2 = 4? Sep 9, 2018 at 15:07
• @user585380 That was a typographical error, which has been fixed. Sep 9, 2018 at 22:56
• @heropup could you look at this problem: math.stackexchange.com/questions/2002756/… Sep 9, 2018 at 22:58
• I up-voted this answer, but I have this qualm about it: This practice of using a random variable as a subscript on the expectation operator is nonsense. It should not be done. Sep 9, 2018 at 23:51

Let $T$ be the number of transactions in a 3 hour interval, $X$ the number of customers in a 3 hour interval (which is $\sim poisson(60)$ as you found), and $N$ the number of transactions per customer (which has expectation 2 as you say and variance 5 as you can verify). Then we have that $T=\sum_{i=1}^X N$. I personally used the conditional variance formula from here, which is a straightforward calculation after this setup.