Prove that if X ∼ Geometric (p) then, E ( X ) = q p Var ⁡ ( X ) = q p 2 m X ( t ) = p ( 1 − q e t )

Question

Prove that if   X  ∼ Geometric (p) then,   E  (  X  )  =      q    p      Var  ⁡  (  X  )  =      q          p              2                    m          X        (  t  )  =  p      (    1    −    q          e              t              )

Belen Solomon · Accepted Answer

Step 1For the variance, it suffices to compute   E  [      X    2    ]. You can adapt the approach you used for computing E[X]; it may help to manipulate the series for       1          (      1      −      x              )        3            .Here is a direct derivation:                    E        [                  X          2                ]                            =                  ∑                      k            ≥            0                                    k          2                p                  q          k                                    (        1        −        q        )        E        [                  X          2                ]        =        E        [                  X          2                ]        −        q        E        [                  X          2                ]                            =                  ∑                      k            ≥            0                                    k          2                p                  q          k                −                  ∑                      m            ≥            0                                    m          2                p                  q                      m            +            1                                                            =                  ∑                      k            ≥            0                                    k          2                p                  q          k                −                  ∑                      k            ≥            1                          (        k        −        1                  )          2                p                  q          k                                                  =                  ∑                      k            ≥            1                          (        2        k        −        1        )        p                  q          k                                                                =        2                  ∑                      k            ≥            1                          k        p                  q          k                −                  ∑                      k            ≥            1                          p                  q          k                                                  =        2        E        [        X        ]        −        (        1        −        p        )        =        2                  q          p                −        q        =                              q            (            2            −            p            )                    p                .            Given the above expression for   (  1  −  q  )  E  [      X    2    ], you can then do a few more steps compute the variance of X.Step 2For the MGF,       m    X    (  t  )  =  E  [      e          t      X        ]  =      ∑          k      ≥      0            e          t      k        p      q    k    =  p      ∑          k      ≥      0        (      e    t    q      )    k   which is a geometric series that you should already know how to compute. By the way, there is a typo in your expression for the MGF.

Austin Rangel · Accepted Answer

Step 1I would do the MGF calculation first, then use the result to compute the expectation and variance. That&#039;s probably not the intent of the problem, but it is mathematically valid.      M    X    (  t  )  =  E  ⁡  [      e          t      X        ]  =      ∑          x      =      0        ∞        e          t      x        (  1  −  p      )    x    p  .Step 2Then,   E  ⁡  [  X  ]  =      M    X    ′    (  0  )  ,    E  ⁡  [      X    2    ]  =      M    X    ″    (  0  )  ,, and   Var  ⁡  [  X  ]  =  E  ⁡  [      X    2    ]  −  E  ⁡  [  X      ]    2    .

Prove that if X∼ Geometric (p) then, E(X)=q/p, Var(X)=q/p^2, m_X(t)=p(1-qe^t).

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