Question

The number of fish that a fisherman catches in a day is a Poisson random variable with mean = 30. However, on average, the fisherman throws back two out of every three fish he catches.

(a) What is the probability that, on a given day, the fisherman takes home n fish.

(b) What is the mean and variance of the number of fish he catches

(c) What is the mean and variance of the number of fish he takes home

(d) What independence assumptions have you made?

Answer 1

Letx be the number of fish that a tisherman. catches in a day X N Poisson (mean=30) ie x=30 P(X=2) = e- xx ; x = 0, 1,2 .. a! (a) probability that, on a given day, the fisherman takes lonce a fisne Let y be the number of fish that the fisherman takes nome

On average, the fisherman throws back two out of every three fish he catches. so ou average of 30 fish we catches; he throws back 20 and take remaining 10 back home Hence were x1=10 PCYzy)= e-nx y = 0, 1, 2... y! Ply=n) - e- 10(10)" 0.000045 X 10" n! (6) The mean and variance of the number of fish we catches = x e. Mean = x=30 and variance = x=30. (Cl The mean & variance of the number of fish we takes nome.al . Mean=21=10 and variance = x1 = 10. (a) Independence assumption - ooo Events occur independent ie catching of a füh does not affect the probability of a catching other fish.