Conditional Probability and Maximum values of random variables including a Geometric Random VariableLet X 1 , X 2 , . . . be i.i.d. random variables with the common CDF F , and suppose they are independent of N, a geometric random variable with parameter p. Let M = m a x { X 1 , . . . , X N }(a) Find P r { M ≤ x } by conditioning on N.(b) Find P r { M ≤ x | N = 1 }.(c) Find P r M ≤ x | N &gt; 1 .Could anyone explain why in (c) P r { M ≤ x | N &gt; 1 } = F ( x ) P r { M ≤ x }

Question

Conditional Probability and Maximum values of random variables including a Geometric Random VariableLet       X    1    ,      X    2    ,  .  .  . be i.i.d. random variables with the common CDF F , and suppose they are independent of N, a geometric random variable with parameter p. Let   M  =  m  a  x  {      X    1    ,  .  .  .  ,      X    N    }(a) Find   P  r  {  M  ≤  x  } by conditioning on N.(b) Find   P  r  {  M  ≤  x      |    N  =  1  }.(c) Find   P  r      M    ≤    x          |        N    &amp;gt;    1  .Could anyone explain why in (c)  P  r  {  M  ≤  x      |    N  &amp;gt;  1  }  =  F  (  x  )  P  r  {  M  ≤  x  }

Bradley Forbes · Accepted Answer

Step 1If you have solved (a) and (b) then you can find the answer of (c) on base of the equality:  P      (    M    ≤    x    )    =  P      (    M    ≤    x    ∣    N    =    1    )    P      (    N    =    1    )    +  P      (    M    ≤    x    ∣    N    &amp;gt;    1    )    P      (    N    &amp;gt;    1    )  Step 2I found   P      (    M    ≤    x    ∣    N    &amp;gt;    1    )    =            p      F                        (          x          )                          2                            1      −              (        1        −        p        )            F              (        x        )            and:   P      (    M    ≤    x    )    =            p      F              (        x        )                    1      −              (        1        −        p        )            F              (        x        )            showing that indeed:  P      (    M    ≤    x    ∣    N    &amp;gt;    1    )    =  F      (    x    )    P      (    M    ≤    x    )

Let X_1, X_2,... be i.i.d. random variables with the common CDF F, and suppose they are independent of N, a geometric random variable with parameter p. Let M=max{X_1,...,X_N}. (a) Find Pr{M le x} by conditioning on N. (b) Find Pr{M le x|N = 1}. (c) Find Pr{M le x|N > 1}.

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