Question

A man named Steven has a virus that spreads very quickly. Steven sees many people each day. The number of people he sees is represented by a Poisson random variable with mean λ. Since Steven didn't stay home and the virus spreads quickly he infects every person he sees with probability w, independently. Let X be the number of people he sees and Y be the number of people he spreads the virus to.

1. Given X = x, what's the conditional probability mass function for Y?
2. What is the joint probability mass function for X and Y?
3. What is the marginal distribution for Y? What is the name and the relevant parameters?

Answer 1

1.

Given X = x, the conditional probability mass function for Y is,

Y | X ~ Binomial(n = X, p = w)

2.

X ~ Poisson(λ)

The joint probability mass function for X and Y is,

3.

The marginal distribution for Y is,

(The range of x is y to infinity)

(Putting t = x-y}

{}

which is Poisson distribution with parameter

Y ~ Poisson()