Conditional expectation of a Geometric random variable conditioned on the event A = { X &gt; 3 }Let X be a geometric random variable and let A denote the event { X &gt; 3 } find the conditional probability mass function of X with respect to the event A and then compute E[X∣A].

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Conditional expectation of a Geometric random variable conditioned on the event   A  =  {  X  &amp;gt;  3  }Let X be a geometric random variable and let A denote the event   {  X  &amp;gt;  3  } find the conditional probability mass function of X with respect to the event A and then compute E[X∣A].

rmercierm7 · Accepted Answer

Step 1We have that   P  (  X  =  k  )  =  p  (  1  −  p      )    k    ,    k  =  0  ,  1  ,  2  ,  …and hence   P  (  X  &amp;gt;  3  )  =  (  1  −  p      )    4  . Now if   k  =  0  ,  1  ,  2  ,  3, then of course   P  (  X  =  k  ∣  X  &amp;gt;  3  )  =  0 and if   k  =  4  ,  5  ,  …, then   P  (  X  =  k  ∣  X  &amp;gt;  3  )  =            P      (      X      =      k      ,      X      &amp;gt;      3      )              P      (      X      &amp;gt;      3      )        =            P      (      X      =      k      )              P      (      X      &amp;gt;      3      )        =  p  (  1  −  p      )          k      −      4        .Step 2The conditional expectation is thus given by   E  [  X  ∣  X  &amp;gt;  3  ]  =      ∑          k      =      4        ∞    k  P  (  X  =  k  ∣  X  &amp;gt;  3  )  =      ∑          k      =      4        ∞    k  p  (  1  −  p      )          k      −      4        =      ∑          k      =      0        ∞    (  k  +  4  )  p  (  1  −  p      )    k    =  4  +  E  [  X  ]  .

Conditional expectation of a Geometric random variable conditioned on the event A={X>3}

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