Poisson distribution cost function

Question

Could I have some assistance with part b of the following question: Is it incorrect to substitute poisscdf(3,3) into the y component of the equation as 3 is the expected value of the distribution?

The number of machine failures per month in a certain plant has a Poisson distribution with mean equal to 3. Present facilities at the plant can repair 4 machines per month. If any additional machines fail then they are repaired by an outside contractor.

1. Use Matlab to find the probability, on a given month, that the contractor is required to work in the plant. Give your answer to three decimal places.
2. The cost (in thousands of dollars) to the plant of machine failures can be approximated as: $$X = 17+5Y+Y^2 $$ Find (to one decimal place) the expected cost of machine failures per month (in thousands of dollars).

Answer 1

HINTS

Let $N$ be the number of failures in a particular month. Then, $N \sim \mathcal{P}(3)$. Do you know what are the expected value and variance of $N$?

1. Can you find $\mathbb{P}[N \ge 5] = 1 - \mathbb{P}[N < 5]$ and relate it to the desired quantity?
2. You are looking for $$\mathbb{E}\left[17 + 5N + N^2\right] = 17 + 5 \mathbb{E}[N] + \mathbb{E}\left[N^2\right].$$ You may want to remember that $$\mathrm{var} N = \mathbb{E}\left[N^2\right] - \mathbb{E}[N]^2...$$