A box contains four green balls. We choose balls at random, with replacement, according to the following rules: (i) Upon choosing each ball from the box, we mark it with a red stripe before replacing it in the box. (ii) We stop as soon as we choose a ball with a red stripe (i.e. the ball has been chosen twice). Let x= the number of times that balls were chosen from the box. (Note that x must be at least 2.) Find the distribution of x.
x is a discrete random variable with mass points 2,3,.....
P(x=2) = P(a ball with a red stripe was chosen on the 2nd choice) = 1/4
P(x=3) = P(a ball with a red stripe was chosen on the 3rd choice) = (3/4)(1/4)
P(x=4) = P(a ball with a red stripe was chosen on the 4th choice) = (3/4)2(1/4)
P(x=5) = P(a ball with a red stripe was chosen on the 5th choice) = (3/4)31/4
Similarly ; P(x=n) = P(a ball with a red stripe was chosen on the n-th choice) = (3/4)n-2(1/4)
Thus ; x has a geometric distribution with pmf :- fx(j) = P(x=j) = (3/4)j-2(1/4)
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