A company provides each of its employees with a death benefit of 100. The company purchases insurance that pays the cost of total death benefits in excess of 400 per year. The number of employees who will die during the year is a Poisson random variable with mean 2. Calculate the expected annual cost to the company of providing the death benefits, excluding the cost of the insurance.
Correct answer: 192
My Solution:
Let X = # of deaths/year ~ Poisson($\lambda=2$)
Let Y = Total cost to company
Then
$$f_Y(y) =
\begin{cases}
0, &P(X=0) \\
100, &P(X=1) \\
200, &P(X=2) \\
300, &P(X=3) \\
400, &P(X=4) \\
100(X-4), &P(X=x), & x \geq 5, x \in \mathbb N
\end{cases}
$$
And we simply calculate $E[Y] = 0 + 100P(X=1) + 200P(X=2) + 300P(X=3)+400P(X=4) + \sum_{x=5}^{\infty} (100x-400)P(X=x) \\= 0 + 100P(X=1) + 200P(X=2) + 300P(X=3)+400P(X=4) + \sum_{x=0}^{\infty} (100x-400)P(X=x) – [(-400)P(X=0) + (-300)P(X=1) + (-200)P(X=2) + (-100)P(X=3)] + 0 \\= constant + E[100X-400] \\ = constant + 100E[X] – 400 \\= constant + 100*(\lambda=2) – 400$
Am I doing this right? I keep getting 178, idk if it's wrong solution or calculation error
Best Answer
I think you have just misread the text: in my opinion, if more then 4 people die, then the company pays 400 and the insurance all the rest. Which makes more sense: the company pays the insurance in order to avoid a large payment in the unexpected case of too many deaths. In my hypothesis, in case of 12 deaths, the company pays only 400, which is more affordable than "yours" (12-4)= 800€
The function should be: $$f_Y(y) = \begin{cases} 0, &P(X=0) \\ 100, &P(X=1) \\ 200, &P(X=2) \\ 300, &P(X=3) \\ 400, &P(X=4) \\ \color{red}{400}, &P(X>4). \end{cases} $$
Hence the evaluation is:
$$ \mathbb E[Y]=0 + 100P(X=1) + 200P(X=2) + 300P(X=3)+400P(X=4) + (400)P(X>4)\\ = 400 P(X\ge 0) - 400P(X=0)-300P(X=1)-200P(X=2)-100 P(X=3) =\\ = 400 - (400+600+400+400/3)e^{-2}\simeq 192,486$$