Geometric probability mass functionGiven two geometric random variables: X j for j = 1 , 2 with probability mass function P ( X 1 = m ) = P ( X 2 = m ) = ( 1 − q ) m − 1 ⋅ q where m = 1 , 2 , … and 0 &lt; q &lt; 1.. Find the probability mass function max ( X 1 , X 2 ) ..First, can we assume that these random variables are independent? If so, is the probability mass function max ( X 1 , X 2 ) equal to the following product max ( ( 1 − q ) 2 m − 2 ⋅ q 2 ) ?

Question

Geometric probability mass functionGiven two geometric random variables:         X          j         for     j  =  1  ,  2   with probability mass function   P  (      X          1        =  m  )  =  P  (      X          2        =  m  )  =  (  1  −  q      )          m      −      1        ⋅  q   where   m  =  1  ,  2  ,  …   and   0  &amp;lt;  q  &amp;lt;  1.. Find the probability mass function         max    (      X          1        ,      X          2        )  ..First, can we assume that these random variables are independent? If so, is the probability mass function         max    (      X          1        ,      X          2        ) equal to the following product       max              (        (  1  −  q      )          2      m      −      2        ⋅      q          2                  )        ?

plomet6a · Accepted Answer

Step 1I&#039;d go this way: say   Z  =      m    a    x    (      X    1    ,      X    2    ).Then for   m  ∈            N              ∗      ,   P  (  Z  =  m  )  =  P  (  (      X    1    =  m        a    n    d          X    2    &amp;lt;  m  )        o    r      (      X    2    =  m        a    n    d          X    1    &amp;lt;  m  )        o    r      (      X    1    =  m        a    n    d          X    2    =  m  )  )Step 2These three events are disjoints, so   P  (  Z  =  m  )  =  P  (  (      X    1    =  m        a    n    d          X    2    &amp;lt;  m  )  )  +  P  (  (      X    2    =  m        a    n    d          X    1    &amp;lt;  m  )  )  +  P  (  (      X    1    =  m        a    n    d          X    2    =  m  )  )Step 3If       X    1   and       X    2   are independant, we have   P  (  Z  =  m  )  =  2  P  (  X  =  m  )  P  (  X  &amp;lt;  m  )  +  P  (  X  =  m      )    2  .The term   P  (  X  &amp;lt;  m  ) is easy to compute (sum of the terms of a geometric sequence)

Given two geometric random variables: X_j for j=1, 2 with probability mass function P(X_1=m)=P(X_2=m)=(1-q)^{m-1} cdot q where m=1,2,… and 0<q<1. Find the probability mass function max(X_1, X_2).

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