# Math Help - Help~ poisson distribution

1. ## Help~ poisson distribution

a field cantains two types of plants. type A is attractive to insects, whereas B is not. The number of insects to be found on a plant of type A can be modelled as a poisson random variable with mean m. no insects will ever be found on a plant B. the proportion of type B plants in the field is p. let X be the number of insects found on a randomly chosen plant.

1)find the probability mass function of X

2)show that the probability generating function of X is given by
π(s)=p+(1-p)*(e^(m*(s-1)). use this result to find the mean and variance of X.

2. Originally Posted by kevek
a field cantains two types of plants. type A is attractive to insects, whereas B is not. The number of insects to be found on a plant of type A can be modelled as a poisson random variable with mean m. no insects will ever be found on a plant B. the proportion of type B plants in the field is p. let X be the number of insects found on a randomly chosen plant.

1)find the probability mass function of X

2)show that the probability generating function of X is given by
π(s)=p+(1-p)*(e^(m*(s-1)). use this result to find the mean and variance of X.
What have you tried so far? Where do you get stuck?

3. Originally Posted by mr fantastic
What have you tried so far? Where do you get stuck?
Hi, in first question, can i seperate pmf in two part (one with p, another with (1-p) )?

4. Originally Posted by kevek
Hi, in first question, can i seperate pmf in two part (one with p, another with (1-p) )?
The pmf of X is defined as follows:

$f(x) = (p)(1) + (1 - p)e^{-m} = p + (1-p) e^{-m}$ for $X = 0$

(since X = 0 always for plant B and sometimes for Plant A),

$f(x) = (p)(0) + (1-p) \frac{m^x e^{-m}}{x!}$ for $X > 0$

(since X is never greater than zero for Plant B but can be for Plant A).

By definition $n(s) = E(s^X)$

$= \left[ p + (1-p) e^{-m}\right] s^0 + (1-p) \sum_{x=1}^{\infty} \frac{m^x e^{-m}}{x!} s^x$

$= p + (1-p) e^{-m} + (1-p) \left[ \sum_{x=0}^{\infty} \frac{m^x e^{-m}}{x!} s^x - e^{-m} \right]$

The sum is just the probability generating function of a Poisson random variable and is well known (and is probably derived somewhere in these forums).

You should know how to get mean and variance from a probability generating function.