Poisson Distribution Mean and Variance

Question

I know that the mean and variance of a Poisson distribution is λ, so I don't understand how the mean and variance in the question are 70 and 700 respectively. Here's question for reference.

I am the most confused with how the variance is a magnitude higher than the mean. Could someone help me understand this?

Answer 1

If the Poisson rate of arrivals is $\lambda = 7$ per hour, and each arrival takes $c = 10$ minutes to serve, then the mean service time is simply $c\lambda = 70$ minutes. I think this should be perfectly intuitive.

Your question is why the variance of the service time is not also equal to $70$ minutes. But the reason is the same as the reason why

$$\operatorname{Var}[cX] = c^2 \operatorname{Var}[X].$$

The $c$ is deterministic. Each customer takes $10$ minutes. The service time variance is therefore $10^2 = 100$ times the variance of the number of customers arriving.

Answer 2

Expanding slightly on the answer by @heropup ...

I'm not certain it is "perfectly intuitive" because it involves knowing what "service time" means. I'm going to guess that is a standard term in the service industry and that it means something like "total minutes of service per hour that are provided by employees". I.e. if it is over 60 minutes you'd better have more than one employee providing the service. So, with that understanding it is now "obvious" that it ought to just be number of customers per hour multiplied by time to serve each one, i.e. $c \lambda$ on average.

The reason for $\textrm{Var} [cX] = c^2 \textrm{Var}[X]$ can be found from the definition of variance.

$$\textrm{Var} [cX] = \frac{\sum_{i=1}^n (cx_i - c\overline{x})^2}{n} = c^2\frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n} = c^2 \textrm{Var}[X].$$

Answer 3

A Poisson distribution is on integers, they have no dimension. No problem to have mean and variance equal-while it is a good question to rise. As you realize correctly, if the random variable $X$ expresses a real multiple of a unit $a$ (like $a= 1$ cm, or $1$ kilo) the mean and the variable are multiple of $a$ and $a^2$ respectively.