Astronomers treat the number of stars in a given volume of space as a Poisson random variable. The density in the Milky Way Galaxy in the vicinity of our solar system is 1 star per 16 cubic light years. (a) What is the probability of 3 or more stars in 16 cubic light years? (b) How many cubic light years of space must be studied so that the probability of one or more stars exceeds 0.92?

a) 0.264

b) 48 cubit light-years

Step-by-step explanation:

For calculating a) What is the probability of 3 or more stars in 16 cubic light years?

a)

λ = 1 star/16 cubic light-years

t = measure t in units of 16 cubic light years.

E (Y) = λ t = (1/16) (16) = 1 star

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

= 1 - [e^-1 + (e^-1) (1) / 1!]

= 0.264

b)

P (X≥1) = 1 - P (X=0)

= 1 - e^-μ

0.95 = 1 - e^-μ

e^-μ = 1 - 0.95

e^-μ = 0.05

ln (e^-μ) = ln (0.05)

-μ = - 3

μ = 3

Therefore 3 x 16 = 48 cubic light years of space