Question

4. Births in a hospital occur randomly at an average rate of 1.8 births per hour. Let X be the number of births in a given hour. (a) Using λ = 1.8, calculate the probability two births in a given hour P(X = 3). (b) What is the probability of observing more than 2 births in a given hour P(X ≥ 2). (hint: the sum of all probabilities is 1.) (c) Calculate E(X) (d) What does this expectation represent? (e) Calculate Var(X) (f) What unique property of the Poisson distribution do your answers to c and e highlight?

Answer 1

a)

P(X=3)=

{e-λ*λx/x!}=

0.1607

b)

P(X>=2)=1-P(X<=1)=

1-∑x=0x e-λ*λx/x!= 1-P(X=0)-P(X=1) =1-0.1653-0.2975 =

0.5372

c)

here mean of distribution=E(x)=λ=

1.8

this tells us that if many random hours have been taken into consideration , then average birth rate per hour in those should be approximately equal to 1.8

d)

Var(X) =λ=1.8

f)

Poisson distribution has mean and variance equal to rate parameter,