Please answer the question legible 1. 20 points An urn initially contains one re

Question

Please answer the question legible

1. 20 points An urn initially contains one red and one blue ball. At each stage, a ball is randomly chosen and then replaced along with another of the same color. Let X denote the selection number of the first chosen ball that is blue. For instance, if the first selection is red and the second blue, then X is equal to 2. (a) 17 points) Find POX a) for a 1. (b) 17 points Show that, with probability 1, a blue ball is eventually chosen, that is, show that P(X oo) 1. (Hint: Note that C (c) 16 points Find ETX]

Explanation / Answer

Initially the urn contain red ball = 1 and blue ball = 1

X = the selection number of the first chosen ball that is blue

X = 1 means first chosen ball is blue and the probablity = 1/2

X = 2 means first chosen ball is red and second chossen ball is blue and the probablity = (1/2)(1/3)

[ initially red = 1 and blue = 1 , P(R) = 1/2 and we replace the red ball along with another red ball and now red = 2 and blue = 1 so P(B) = 1/3 so P(RB) = (1/2)(1/3) ]

X = 3 means  first two chosen ball is red and third chossen ball is blue and the probablity = (1/2)(2/3)(1/4) = (1/3)(1/4)

[ initially red = 1 and blue = 1 , P(R) = 1/2 and we replace the red ball along with another red ball and now red = 2 and blue = 1 , so P(R) = 2/3 and again we replace the red ball along with another red ball and now red = 3 and blue = 1 so P(RRB) = (1/2)(1/3)(1/4) = (1/3)(1/4) ]

and X = 4, 5, ......so on

so the distribution of X

[ here 2.3 means 2 * 3 , 3.4 = 3 * 4]

(a) P(X >= a) = P(X= a) + P(X = a+1) + P(X = a+2) + P(X = a+3) + ..........................

=1/a(a+1) + 1/(a+1)(a+2) + 1/(a+2)(a+3) + 1/(a+3)(a+4) + .........................................

= 1/a

(b)P(X < infinity) = P(X=1) + P(X=2) + P(X=3) + P(X=4) + ................

= 1/2 + 1/2.3 + 1/3.4 + 1/4.5 + ............

= 1

(c) E[X] = (1 * 1/2) + (2 * 1/2.3) + (3 * 1/3.4) + (4 * 1/4.5) + ..........

= 1/2 + 1/3 + 1/4 + 1/5 + ..............

=

X 1 2 3 4 5 ............... fi 1/2 1/2.3 1/3.4 1/4.5 1/5.6 ...............

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