X and Y are geometric RV's with parameter p.
A) $P {X + Y = n} (n = 1, 2, . . .)$ ?

Question

X and Y are geometric RV&#039;s with parameter p.A)   P  {  X  +  Y  =  n  }  (  n  =  1  ,  2  ,  .  .  .  )?

Cael Cox · Accepted Answer

Step 1I am assuming X and Y are independent. Without this assumption, it is not possible to answer your question without further information.Step 2So, you have to be a little more clever: Suppose we let   Z  =  X  +  Y. We wish to find   Pr  [  Z  =  n  ]. Note that we can write   Pr  [  Z  =  n  ]  =      ∑          y      =      0        n    Pr  [  (  X  =  n  −  y  )  ∩  (  Y  =  y  )  ]  =      ∑          y      =      0        n    Pr  [  X  =  n  −  y  ]  Pr  [  Y  =  y  ]  .Now you know what each of these probabilities is, so write them out, and calculate the sum.

Kade Reese · Accepted Answer

Step 1For the shifted geometric distribution, of X failures before success on trial   X  +  1,   X  ∼      S    G    e    o    (  p  )    ⟺    Pr  (  X  =  x  )  =  p  (  1  −  p      )    x                      Given:                     X        ⊥        Y        ,        X        ∼                  S          G          e          o                (        p        )        ,        Y        ∼                  S          G          e          o                (        p        )                            Pr        (        X        +        Y        =        n        )                            =        Pr                  (                      ⋃                          x              =              0                                      n                                (          X          =          x          ∩          Y          =          n          −          x          )          )                            ∀        n        ∈        {        0..        ∞        }                                                         =                  ∑                      x            =            0                                n                          Pr        (        X        =        x        )        Pr        (        Y        =        n        −        x        )                                                         =                  ∑                      x            =            0                                n                          p        (        1        −        p                  )          x                ⋅        p        (        1        −        p                  )                      n            −            x                                                                           =        (        n        +        1        )                  p          2                (        1        −        p                  )          n                                                                                      Pr        (        X        =        k        ∣        X        +        Y        =        n        )                            =                                            Pr              (              X              =              k              ∩              X              +              Y              =              n              )                                      Pr              (              X              +              Y              =              n              )                                                  ∀        n        ∈        {        0..        ∞        }        ,        k        ∈        {        0..        n        }                                                         =        et cetera            Step 2For the geometric distribution of   X  −  1 failures before success on trial X,  X  ∼      G    e    o    (  p  )    ⟺    Pr  (  X  =  x  )  =  p  (  1  −  p      )          x      −      1                          Given:                     X        ⊥        Y        ,        X        ∼                  G          e          o                (        p        )        ,        Y        ∼                  G          e          o                (        p        )                            Pr        (        X        +        Y        =        n        )                            =        Pr                  (                      ⋃                          x              =              1                                      n              −              1                                (          X          =          x          ∩          Y          =          n          −          x          )          )                            ∀        n        ∈        {        2..        ∞        }                                                         =                  ∑                      x            =            1                                n            −            1                          Pr        (        X        =        x        )        Pr        (        Y        =        n        −        x        )                                                         =                  ∑                      x            =            1                                n            −            1                          p        (        1        −        p                  )                      x            −            1                          ⋅        p        (        1        −        p                  )                      n            −            x            −            1                                                                           =        (        n        −        1        )                  p          2                (        1        −        p                  )                      n            −            2                                                                                                Pr        (        X        =        k        ∣        X        +        Y        =        n        )                            =                                            Pr              (              X              =              k              ∩              X              +              Y              =              n              )                                      Pr              (              X              +              Y              =              n              )                                                  ∀        n        ∈        {        2..        ∞        }        ,        k        ∈        {        1..        n        −        1        }                                                         =        et cetera

X and Y are geometric RV's with parameter p. A) P{X+Y=n}(n=1,2,...) ?

Answered question

Answer & Explanation

New Questions in High school geometry