Question

6-4. find p(x = 4) if x has a poisson distribution such that 3p(x = 1) = p(x = 2)

Answer 1

A random variable
`X` following a Poisson distribution with rate parameter
`\lambda` has PMF


`f_X(x)=(e^(-\lambda)\lambda^x)/(x!)`

We're given that
`3\mathbb P(X=1)=\mathbb P(X=2)`, so


`(3e^(-\lambda)\lambda^1)/(1!)=(e^(-\lambda)\lambda^2)/(2!)`

`\implies3\lambda=\frac{\lambda^2}2`

`\implies\lambda^2-6\lambda=\lambda(\lambda-6)=0`


`\lambda` must be positive, so we arrive at
`\lambda=6`, which means


`\mathbb P(X=4)=(e^(-6)6^4)/(4!)=(54)/(e^6)\approx0.1339`