Chapter 7: Q7.35 (page 361)

The probability generating function of the discrete non-negative integer-valued random variable $X$ having probability mass function $p_{j}, j \geq 0$ is defined by
$ϕ (s) = E [s^{X}] = \sum_{j = 0}^{\infty} p_{j} s^{j}$
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
$ϕ (s) = P {X < Y}$

Short Answer

Expert verified

Hence, the statement $ϕ (s) = \sum_{j = 0}^{\infty} p_{j} s^{j}$ 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,
$ϕ (s) = \sum_{j = 0}^{\infty} p_{j} s^{j}$

:Explanation

The probability generating function of the discrete nonnegative integer values random variable $X$ having the probability mass function $p_{j}$ is,
$ϕ (s) = \sum_{j = 0}^{\infty} p_{j} s^{j}$

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

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

$G_{Z} (z) = \sum_{k = 1}^{\infty} (1 - p)^{k - 1} p z^{k}$

$= p z \sum_{k = 0}^{\infty} [(1 - p) z]^{k}$

$= \frac{p z}{[1 - (1 - p z)]}$

:Explanation

Show that
$ϕ (s) = P (X < Y)$

= P (Y > X)

Explanation

Now,
$P (Y > X) = \sum_{j} P (Y > X ∣ X = j) p_{j}$

$= \sum_{j} P (Y > j ∣ X = j) p_{j}$

$= \sum_{j} P (Y > j) p_{j}$

$= \sum_{j} P (1 - p)^{j} p_{j}$

$= \sum_{j = 0}^{\infty} p_{j} s^{j}$

:Final Answer

Hence, the statement $ϕ (s) = \sum_{j = 0}^{\infty} p_{j} s^{j}$ is proved.

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