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).

mydaruma25 2022-09-27 Answered
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)."
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Answers (1)

altaryjny94
Answered 2022-09-28 Author has 14 answers
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 2
Now recall the definition of the generating function:
π x ( q ) = k = 0 P ( X = k ) q k .
Step 3
By unicity of generating functions you gain P ( X = k ) = 1 3 2 k 1 if k 1, and P ( X = 0 ) = 1 3 .
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