Question

Suppose R is a binomial random variable with r=2 trials and success probability p=0.5. Let A and B be uniform random variables on [0,1] and that A, B, R are mutually independent. Find PDF for Z=RAB. (Solve P(Z<=z) for z ∈[0,1] by conditioning on the value of R ∈{0,1,2}.)

Answer 1

Given :-

R is a binomial random variable with r=2 trials and success probability p=0.5
So, FR(x) = rCx.px.(1-p)r-x =  2Cx*0.5x0.52-x = 2Cx*0.52 = 0.25.(2!/x!.(2-x)!) where x {0,1,2}

A and B are uniform random variables on [0,1]
So, FA(a) = FB(b) = 1 for a,b [0, 1]

Z = RAB
So,

where k {0,1,2}

If k = 0, then

Similarly if k= 1, then

and if k = 2, then

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