How can you find P ( X Y − X &lt; 0 ) if X ∼ G e o m e t r i c ( p ) and Y ∼ B e r n o u l l i ( p )Let the independent random variables X ∼ G e o m e t r i c ( p ) and Y ∼ B e r n o u l l i ( p ), I want to prove that P ( X Y − X &lt; 0 ) = ( p − 1 ) 2 ( p + 1 )Do I need the joint probability mass function for this or should it be proven in some other way?

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How can you find   P  (      X          Y      −      X        &amp;lt;  0  ) if   X  ∼  G  e  o  m  e  t  r  i  c  (  p  ) and   Y  ∼  B  e  r  n  o  u  l  l  i  (  p  )Let the independent random variables   X  ∼  G  e  o  m  e  t  r  i  c  (  p  ) and   Y  ∼  B  e  r  n  o  u  l  l  i  (  p  ), I want to prove that   P  (            X              Y        −        X              &amp;lt;  0  )  =  (  p  −  1      )          2        (  p  +  1  )Do I need the joint probability mass function for this or should it be proven in some other way?

Octavio Barr · Accepted Answer

Step 1Given that Y is Bernoulli distributed, it will have probability mass function   P  (  Y  =  0  )  =  (  1  −  p  ) and   P  (  Y  =  1  )  =  p. As Frank has helpfully commented, we can condition on Y.When   Y  =  0, we have   P  (      X          Y      −      X        &amp;lt;  0      |    Y  =  0  )  =  P  (  −  X  &amp;lt;  0      |    Y  =  0  )  =  P  (  X  &amp;gt;  0      |    Y  =  0  )When   Y  =  1, we   P  (      X          Y      −      X        &amp;lt;  0      |    Y  =  1  )  =  P  (      X          1      −      X        &amp;lt;  0      |    Y  =  1  )  =  P  (  X  &amp;gt;  1      |    Y  =  1  )Step 2Thus, we have the following (the third line being a consequence of the independence between X and Y)                    P                  (                      X                          Y              −              X                                &amp;lt;          0          )                                    =        P                  (                      X                          Y              −              X                                &amp;lt;          0                      |                    Y          =          0          )                P        (        Y        =        0        )        +        P                  (                      X                          Y              −              X                                &amp;lt;          0                      |                    Y          =          1          )                P        (        Y        =        1        )                                          =        P        (        X        &amp;gt;        0                  |                Y        =        0        )        P        (        Y        =        0        )        +        P        (        X        &amp;gt;        1                  |                Y        =        1        )        P        (        Y        =        1        )                                          =        P        (        X        &amp;gt;        0        )        P        (        Y        =        0        )        +        P        (        X        &amp;gt;        1        )        P        (        Y        =        1        )                                          =                  (                      ∑                          k              =              1                                      ∞                                p          (          1          −          p                      )            k                    )                (        1        −        p        )        +                  (                      ∑                          k              =              2                                      ∞                                p          (          1          −          p                      )            k                    )                (        p        )                                          =        (        1        −        p                  )          2                +        p        (        1        −        p                  )          2                                                  =        (        p        −        1                  )          2                (        1        +        p        )

How can you find P(X/(Y-X)<0) if X∼Geometric(p) and Y∼Bernoulli(p)

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