# Thread: geometric and Poisson distributions

1. ## geometric and Poisson distributions

I got the following two questions which I have no clue how to do it.

(1)There are three coins in a barrel. These coins when flipped will come up heads with respective probabilities 0.3, 0.5 and 0.7. A coin is randomly selected from among these three coins and is then flipped until a head appears. Let N denote the number of flips necessary.

What is the probability that N = n, n = 1, 2, 3, . . .?
Is N a geometric random variable?

Suppose now that each time a coin is flipped, it is returned to the barrel and a coin is then selected at random from the three coins, what is the distribution of N?

(2) The number of fish that a fisherman catches in a day is a Poisson random variable with mean 30. On average the fisherman throws back two out of every three fish he catches.

Find the probability that, on a given day, the fisherman takes home k fish, stating any independence assumptions that you make.

Can anyone give me some hints please?

Thank you so much

2. Originally Posted by dangkhoa
I got the following two questions which I have no clue how to do it.

(1)There are three coins in a barrel. These coins when flipped will come up heads with respective probabilities 0.3, 0.5 and 0.7. A coin is randomly selected from among these three coins and is then flipped until a head appears. Let N denote the number of flips necessary.

What is the probability that N = n, n = 1, 2, 3, . . .?
Is N a geometric random variable?
Have you worked out $P(N=n)$ for this situation? If so is there a $p$ such that

$P(N=n)=(1-p)^n p$?

Which is what would be required for $N$ to have a Geometric distribution.

Suppose now that each time a coin is flipped, it is returned to the barrel and a coin is then selected at random from the three coins, what is the distribution of N?
Now the probability of a head on each toss is the same so $N$ has a geometric distribution. All you need do is calculate the probability of a head on a single toss.

CB

3. Originally Posted by dangkhoa
(2) The number of fish that a fisherman catches in a day is a Poisson random variable with mean 30. On average the fisherman throws back two out of every three fish he catches.

Find the probability that, on a given day, the fisherman takes home k fish, stating any independence assumptions that you make.

Can anyone give me some hints please?

Thank you so much
Assume that each fish is thrown back with probability 1/3 independently of what happens to any other fish.

Then the number he takes home will have a Poisson distribution with mean 10.

CB