Theorem 7-10. If X is a random variable which assumes only integral values with probability distribution : P (X = k) = Pr; k = 0, 1, 2, ... and P (X > k) = qk, k 2 0 so that %3D k 9k = Pk+ 1+ Pk+2 + ... = 1-2 pi, and two generating functions are : i=0 P (s) = Po + P1s + P2 s² + ... Q (s) = 90 + 91 s + 92 s² + .. and then for -1

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Prove the theorem

Theorem 7-10. If X is a random variable which assumes only integral values with
probability distribution :
P (X = k) = Pr; k = 0, 1, 2, ... and P (X> k) = qk, k 2 0 so that
%3D
2H.A
9k = Pk+ 1+ Pk+2 + ... = 1-2 p;, and two generating functions are :
k
%3D
i=0
P (s) = Po + Pis + P2 s² +
and
Q (s) = qo + 91 s + 92 s2 + ..
...
Q (s) = 1- P (s)
1-s
then for -1<s<1,
|
%3D
Transcribed Image Text:Theorem 7-10. If X is a random variable which assumes only integral values with probability distribution : P (X = k) = Pr; k = 0, 1, 2, ... and P (X> k) = qk, k 2 0 so that %3D 2H.A 9k = Pk+ 1+ Pk+2 + ... = 1-2 p;, and two generating functions are : k %3D i=0 P (s) = Po + Pis + P2 s² + and Q (s) = qo + 91 s + 92 s2 + .. ... Q (s) = 1- P (s) 1-s then for -1<s<1, | %3D
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