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30 October, 12:43

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?

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  1. 30 October, 14:36
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    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
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