 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/5/a/solution

k = 1

f_{X} (x) = e^{- x}

for

x > 0

and zero for

x < 0

; there is no atom at zero. This is an exponential distribution of rate one. Its first two moments are

\int_{0}^{\infty} x e^{- x} d x = 1

and

\int_{0}^{\infty} x^{2} e^{- x} d x = 2

, so

E [X] = 1, Var (X) = 2 - 1^{2} = 1.

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