Question

8. General Functions of Random Variables. a) Timothy will fip a fair coin until he gets a Tails. Lt X be the number of lips, and let X2 be the number of points Timothy scores. Find the PMF of Y, and find P(Y> 10). b) Jimothy will flip a fair coin 5 times. Let X be the number of times he gets Tails, and let Y -2* be the number of points that Jimothy scores. Find the PMF of Y and find P(Y Is Timothy or Jimothy more likely to score more than 10 points? 10). c) Challenge: Let X ~ Erp(A) and let Y = 1-e-Ar. Show that Y ~ U(0.1). Note: For more information on this problem, google the "probability integral transformation" or ask me for details

Answer 1

(a) X: number of flips till he gets a tails

P(X=1)= P(tails in one toss of a coin) =1/2

P(X=2) = P(heads in 1^st toss, tails in 2^nd ) = (1/2)*(1/2)=(1/2)²

P(X=3) = P(heads in 1^st and 2^nd toss, tails in 3^rd toss)= (1/2)²*(1/2)=(1/2)³

And so on..

And Y = X²

Hence, we obtain the pmf of X and Y as follows

X	1	2	3	4	So on
Y	1	4	9	16	So on
Probability	1/2	(1/2)2	(1/2)3	(1/2)4	So on

P( Y >10) = 1- P( Y ? 10) = 1- (P( Y =1) + P(Y= 4) + P( Y=9))

                  = 1- (7/8)

                  = 1/8 = 0.125

(b)  

times he gets tai us İ.e· XA, B(5,1/2. no 32 -

jimothy is more likely to score more than 10 points than timothy since 0.125<0.1875.

(c)

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