A certain discrete random variable has probability generating function: π x ( q ) = 1 3 2 + q 2 − q Compute p ( x ) for x = 0 , 1 , 2 , 3 , 4 , 5. (Hint: the formula for summing a geometric series will help you expand the denominator)."

Question

A certain discrete random variable has probability generating function:       π    x    (  q  )  =            1      3                          2        +        q                    2        −        q            Compute   p  (  x  )     for     x  =  0  ,  1  ,  2  ,  3  ,  4  ,  5. (Hint: the formula for summing a geometric series will help you expand the denominator).&quot;

altaryjny94 · Accepted Answer

Step 1                              π          x                (        q        )                            =                  1          3                (        2        +        q        )                  (                      1            2                                1                          1              −                              (                                  q                  2                                )                                              )                                                  =                              2            +            q                    6                          (                      ∑                          k              =              0                                      ∞                                                          (                              q                2                            )                        k                    )                                                  =                  ∑                      k            =            0                                ∞                                    1                      3            ⋅                          2              k                                                q          k                +                  ∑                      k            =            0                                ∞                                    1                      3            ⋅                          2                              k                +                1                                                              q                      k            +            1                                                            =                  1          3                +                  ∑                      k            =            1                                ∞                                    1                      3            ⋅                          2              k                                                q          k                +                  ∑                      k            =            1                                ∞                                    1                      3            ⋅                          2              k                                                q          k                                                  =                  1          3                +                  ∑                      k            =            1                                ∞                                    1                      3            ⋅                          2                              k                −                1                                                              q          k                .            Step 2Now recall the definition of the generating function:      π    x    (  q  )  =      ∑          k      =      0              ∞            P    (  X  =  k  )      q    k    .Step 3By unicity of generating functions you gain       P    (  X  =  k  )  =      1          3      ⋅              2                  k          −          1                     if   k  ≥  1, and       P    (  X  =  0  )  =      1    3  .

A certain discrete random variable has probability generating function: pi_x(q)=1/2 (2+q)/(2-1). Compute p(x) for x = 0,1,2,3,4,5. (Hint: the formula for summing a geometric series will help you expand the denominator).

Answered question

Answer & Explanation

New Questions in High school geometry