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February 27, 2023February 27, 2023 Discipline: Statistics
Learning Goal: I’m working on a statistics test / quiz prep and need a reference to help me learn.

Example exercise: Let X be a discrete random variable uniformly distributed on { 1 , 2 } , that is P ( X = x ) = 1/2, x = 1 , 2 .

Let Y be another random variable and assume that the conditional distribution of Y given X = x is exponential with parameter x (i.e. the p.d.f. of Y | X = x is f Y | X = x ( y ) = xe^-(xy) , y>= 0 ).

Then the unconditioned probability that Y is > 1 is

(a) P ( Y > 1) = 1/2 ( e^-x + e^-2 x )

(b) P ( Y > 1) = 1/2 ( e^-1 + e^-2 )

(c) P ( Y > 1) = 1-1/2 ( e^-1 + e^-2 )

(d) P ( Y > 1) = 1/2

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