Conditional probability involving a geometric random variableLet X 1 , . . . be independent random variables with the common distribution function F, and suppose they are independent of N, a geometric random variable with parameter p. Let M = m a x ( X 1 , . . . , X N ). Find P ( M ≤ x ) by conditioning on N.

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Conditional probability involving a geometric random variableLet       X    1    ,  .  .  . be independent random variables with the common distribution function F, and suppose they are independent of N, a geometric random variable with parameter p. Let   M  =  m  a  x  (      X    1    ,  .  .  .  ,      X    N    ). Find   P  (  M  ≤  x  ) by conditioning on N.

Paige Paul · Accepted Answer

Step 1The problem statement says that       X    1    ,      X    2    ,  .  .  . all have the same distribution.Now if I understand the problem statement correctly, you should find   P  (  M  ≤  x  ), however it seems to be easier to calculate   P  (  M  ≤  x      |    N  =  k  ) and then calculate   P  (  M  ≤  x  ).Step 2Note that:   P  (  M  ≤  x  )  =      ∑          k      =      1              ∞        P  (  M  ≤  x  ,  N  =  k  )  =      ∑          k      =      1              ∞        P  (  N  =  k  )  P  (  M  ≤  x      |    N  =  k  )

Let X_1,... be independent random variables with the common distribution function F, and suppose they are independent of N, a geometric random variable with parameter p. Let M=max(X_1,...,X_N). Find P(M le x) by conditioning on N.

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