Consider independent trials, each of which results inoutcome i,i  =  0, 1,... ,k, with pro...

Consider independent trials, each of which results in

outcome i,i  =  0, 1,... ,k, with probability pi,

Let N denote the number of trials needed to obtain an outcome that is not equal to 0, and let X be that outcome.

(a) Find P{N  = n}, n > 1.

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(b) Find P{X  =  j}, j =1,.., k.

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(c) Show that P{N = n, X = j} = P{N = n}P{X = j}.

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(d) Is it intuitive to you that N is independent of X?

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(e) Is it intuitive to you that X is independent of N?