Statistics

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 one star per 16 cubic light years.
(a) What is the probability of three 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 0.95?

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  1. a) 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)(16) = 48 cubic light years of space

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