Memoryless property and geometric distributionSuppose X is a random variable taking values in N 0 with the memoryless property,i.e., for each pair of number s , t ∈ N , P ( X ≥ s + t ∣ X &gt; t ) = P ( X ≥ s )Show that a random variable X with values in N 0 has the memoryless property if and only if X ∼ Geometric ( p ) of parameter p = P ( X = 1 ).

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Memoryless property and geometric distributionSuppose X is a random variable taking values in             N        0   with the memoryless property,i.e., for each pair of number   s  ,  t  ∈      N  ,  P  (  X  ≥  s  +  t  ∣  X  &amp;gt;  t  )  =  P  (  X  ≥  s  )Show that a random variable X with values in             N        0   has the memoryless property if and only if   X  ∼  Geometric  (  p  ) of parameter   p  =  P  (  X  =  1  ).

Monastro3n · Accepted Answer

Step 1We have   P  (  X  ≥  s  +  t  ∣  X  &amp;gt;  t  )  =  P  (  X  ≥  s  ) and   P  (  X  ≥  s  +  t  ∣  X  &amp;gt;  t  )  =                    P        (        X        ≥        s        +        t        )                    P        (        X        &amp;gt;        t        )             therefore   P  (  X  ≥  s  +  t  )  =  P  (  X  ≥  s  )  P  (  X  &amp;gt;  t  )  .Summing the last equality once over s then once over t we obtain   I  :      ∑          s      =      1              ∞        P  (  X  ≥  s  +  t  )  =      ∑          s      =      1              ∞        P  (  X  ≥  s  )  P  (  X  &amp;gt;  t  )  I  I  :      ∑          t      =      1              ∞        P  (  X  ≥  s  +  t  )  =      ∑          t      =      1              ∞        P  (  X  ≥  s  )  P  (  X  &amp;gt;  t  )Now in I set   t  =  k, and set   s  =  k in II so that their left hand side be equal and consequently  P  (  X  ≥  k  )      ∑          t      =      2              ∞        P  (  X  ≥  t  )  =  P  (  X  ≥  k  +  1  )      ∑          t      =      1              ∞        P  (  X  ≥  t  )  .Step 2Now let   α  =      ∑          t      =      2              ∞        P  (  X  ≥  t  ) and evaluate for   k  =  1 to obtain,  α  =            1      −      p        p    ,where   p  =  P  (  X  =  1  )  .. Therefore   (  1  −  p  )  P  (  X  ≥  k  )  =  P  (  X  ≥  k  +  1  ). Since   P  (  X  ≥  1  )  =  1 we have that   P  (  X  ≥  k  )  =  (  1  −  p      )          k      −      1        ;  k  =  1  ,  2  ,  .  .  .implying that   P  (  X  =  k  )  =  (  1  −  p      )          k      −      1        −  (  1  −  p      )          k        =  p  (  1  −  p      )          k      −      1        .Therefore   X  ∼ Geometric(p).

Suppose X is a random variable taking values in N_0 with the memoryless property,i.e., for each pair of number s, t in N, P(X ge s+t∣X>t)=P(X ge s). Show that a random variable X with values in N_0 has the memoryless property if and only if X∼Geometric(p) of parameter p=P(X=1).

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