Question

The probability generating function of the discrete non-negative integer-valued random variable $X$having probability mass function $p_j,j\ge 0$is defined by

$\varphi \left(s\right)=E\left[s^X\right]=\sum _{j=0}^{\infty }{p}_j{s}^j$

Let $Y$be a geometric random variable with parameter $p=1-s$, where $0<s<1$. Suppose that $Y$is independent of $X$, and show that
$\varphi \left(s\right)=P\{X<Y\}$

Short answer

Hence, the statement $\varphi \left(s\right)=\sum _{j=0}^{\infty }{p}_j{s}^j$is proved.

Step-by-step solution

Concept Introduction

The probability generating function of the discrete nonnegative integer values random variable $X$having the probability mass function $p_j$ is,
$\varphi \left(s\right)=\sum _{j=0}^{\infty }{p}_j{s}^j$

:Explanation

The probability generating function of the discrete nonnegative integer values random variable $X$having the probability mass function $p_j$ is,
$\varphi \left(s\right)=\sum _{j=0}^{\infty }{p}_j{s}^j$

Let $Y$be a geometric random variable with parameter $p=1-s$.

Then $p_k:=(1-p{)}^{k-1}p$

$G_Z\left(z\right)=\sum _{k=1}^{\infty }(1-p{)}^{k-1}p{z}^k$

$=pz\sum _{k=0}^{\infty }\left[\right(1-p)z{]}^k$

$=\frac{pz}{[1-(1-pz\left)\right]}$

:Explanation

Show that
$\varphi \left(s\right)=P(X<Y)$

$=P(Y>X)$

Explanation

Now,
$P(Y>X)=\sum _{j}P(Y>X\mid X=j)p_j$

$=\sum _{j}P(Y>j\mid X=j)p_j$

$=\sum _{j}P(Y>j)p_j$

$=\sum _{j}P(1-p{)}^j{p}_j$

$=\sum _{j=0}^{\infty }{p}_j{s}^j$

:Final Answer

Hence, the statement $\varphi \left(s\right)=\sum _{j=0}^{\infty }{p}_j{s}^j$is proved.