Find the probability that a geometric random variable X is an even numberLet α be the probability that a geometric random variable X with parameter p is an even numbera) Find α using the identity α = ∑ i = 1 ∞ P [ X = 2 i ].b)Find α by conditioning on wether X = 1 or X &gt; 1

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Find the probability that a geometric random variable X is an even numberLet   α be the probability that a geometric random variable X with parameter p is an even numbera) Find   α using the identity   α  =      ∑          i      =      1              ∞        P  [  X  =  2  i  ].b)Find   α by conditioning on wether   X  =  1 or   X  &amp;gt;  1

esbalatzaj · Accepted Answer

Step 1We have                     P        (                              X                     is even                )                    =        P        (                              X                     is even                ∣        X        =        1        )        P        (        X        =        1        )        +        P        (                              X                     is even                ∣        X        &amp;gt;        1        )        P        (        X        &amp;gt;        1        )                                  =        P        (                              X                     is even                ∣        X        &amp;gt;        1        )        P        (        X        &amp;gt;        1        )             since obviously the first term on the RHS is zero. Now the probability that X is even, given that the first trial was a failure, is the same as the probability that X is odd (because after the first failure you have an identical experiment, but you need an odd result in this experiment in order to get an even result overall).Step 2So   α  =  (  1  −  α  )  (  1  −  p  ) and solving gives   α  =            1      −      p              2      −      p           .

skilpadw3 · Accepted Answer

Step 1Let   α be the required probability. In order for X to be even, we must have a failure on the first trial, and have the number of trials after the first be odd. The probability of that, given that the first trial was a failure, is   1  −  α.Step 2Thus   α  =  (  1  −  p  )  (  1  −  α  )  .. Solve for   α.

Let alpha be the probability that a geometric random variable X with parameter p is an even number. a) Find alpha using the identity alpha=sum_{i=1}^{infty}P[X=2i]. b)Find alpha by conditioning on wether X=1 or X>1.

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