In: Statistics and Probability
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}.)
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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