Question

6-1. let x have a poisson distribution with a mean of 4. find (a) p(2≤x≤5). (b) p(x≥3). (c) p(x≤3).

Answer 1

The mean of a Poisson distribution coincides with its rate parameter, so
`\lambda=4`. We have a PMF of


`f_X(x)=(e^(-4)4^x)/(x!)`

So,

a.
`\mathbb P(2\le X\le5)=(e^(-4)4^2)/(2!)+\cdots+(e^(-4)4^5)/(5!)\approx0.6936`

b.
`\mathbb P(X\ge3)=(e^(-4)4^3)/(3!)+(e^(-4)4^4)/(4!)+\cdots\approx0.7619`

c.
`\mathbb P(X\le3)=(e^(-4)4^0)/(0!)+\cdots+(e^(-4)4^3)/(3!)\approx0.4335`