Question

The random variable X follows a Poisson process with the given mean. Assuming mu equals 7, compute the following.

​(a) ​P(4​) ​

(b) ​P(X<4​) ​

(c) ​P(Xgreater than or equals4​) ​

(d) ​P(4less than or equalsXless than or equals8​)

Answer 1

a)

Here, λ = 7 and x = 4
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X = 4)
P(X = 4) = 7^4 * e^-7/4!
P(X = 4) = 0.0912
Ans: 0.0912

b)

Here, λ = 7 and x = 4
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X < 4).
P(X < 4) = (7^0 * e^-7/0!) + (7^1 * e^-7/1!) + (7^2 * e^-7/2!) + (7^3 * e^-7/3!)
P(X < 4) = 0.0009 + 0.0064 + 0.0223 + 0.0521
P(X < 4) = 0.0817

c)

Here, λ = 7 and x = 4
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X > =4).
P(X <= 3) = (7^0 * e^-7/0!) + (7^1 * e^-7/1!) + (7^2 * e^-7/2!) + (7^3 * e^-7/3!)
P(X <= 3) = 0.0009 + 0.0064 + 0.0223 + 0.0521
P(X <= 3) = 0.0817

P(X> =4) = 1- P(x< =3)
= 1- 0.0817
= 0.9183

d)

Here, λ = 7, x1 = 4 and x2 = 8.
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(4 <= X <= 8)
P(4 <= X <= 8) = (7^4 * e^-7/4!) + (7^5 * e^-7/5!) + (7^6 * e^-7/6!) + (7^7 * e^-7/7!) + (7^8 * e^-7/8!)
P(4 <= X <= 8) = 0.0912 + 0.1277 + 0.149 + 0.149 + 0.1304
P(4 <= X <= 8) = 0.6473