What is the distribution of min of two independent geometric rv?Let X and Y be independent discrete random variables, X having the geometric distribution with parameter p and Y having the geometric distribution with parameter r. Show that U = m i n { X , Y } has the geometric distribution with parameter p + r − p r

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What is the distribution of min of two independent geometric rv?Let X and Y be independent discrete random variables, X having the geometric distribution with parameter p and Y having the geometric distribution with parameter r. Show that   U  =  m  i  n  {  X  ,  Y  } has the geometric distribution with parameter   p  +  r  −  p  r

Raelynn Johnson · Accepted Answer

Explanation:To have   X  ≥  x attempts you need to have   x  −  1 failures with probability   (  1  −  p      )          x      −      1       so use something like  P  (  U  ≥  u  )  =  P  (  X  ≥  u  ,  Y  ≥  u  )  =  P  (  X  ≥  u  )  P  (  Y  ≥  u  )    =  (  1  −  p      )          u      −      1        (  1  −  r      )          u      −      1        =            (        (  1  −  p  )  (  1  −  r  )                    )                    u      −      1          =  (  1  −  (  p  +  r  −  p  r  )      )          u      −      1

Let X and Y be independent discrete random variables, X having the geometric distribution with parameter p and Y having the geometric distribution with parameter r. Show that U=min{X,Y} has the geometric distribution with parameter p+r-pr

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