Question

You have three independent uniform random variables Xi on [0,1] for i=1,2,3. calculate

(a) P(all of them are less than 1/2)

(b) P(at least one of them is less than 1/2)

(c) the conditional probability P(all of them are less than 1/2 | at least one of them is less than 1/2)

(d) the mean and the variance of S = X1 + X2 + X3

(e) P(the value of X2 lies between the values of the other two random variables)

Answer 1

P(X <x) = x

(a) P(all of them are less than 1/2)

=P(X1 < 1/2 , X2 < 1/2 , X3 < 1/2)

= P(X1 < 1/2 ) *P(X2 < 1/2) P( X3 < 1/2)

=1/2^3 = 1/8

(b) P(at least one of them is less than 1/2)

= 1 - P(none of them is less than 1/2)

= 1- (1/2)^3

=7/8

(c) the conditional probability P(all of them are less than 1/2 | at least one of them is less than 1/2)

= P(X1 < 1/2 , X2 < 1/2 , X3 < 1/2) | at least one of them is less than 1/2)

= P(X1 < 1/2 , X2 < 1/2 , X3 < 1/2) and at least one of them is less than 1/2) /P(at least one of them is less than 1/2)

= P(X1 < 1/2 , X2 < 1/2 , X3 < 1/2)/( P(X1 < 1/2 , X2 < 1/2 , X3 < 1/2))

= 1/7

(d) the mean and the variance of S = X1 + X2 + X3

mean (S) = mean (X1 + X2 + X3 ) = 3 *1/2 = 3/2

variance(S) = 3 var(X) = 3 * 1/12 = 1/4

e)

since Xi are independent ,

we can arrange X1,x2 and X3 is 3! = 6

number of ways X2 is in between = 2

hence required probability = 2/6 = 1/ 3

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