Chapter 7: Q.7.4 (page 362)

Use the conditional variance formula to determine the variance of a geometric random variable $X$ having parameter $p$ .

Short Answer

Expert verified

The variance of a geometric random variable $X$ having parameter $p$ is $Var (X) = \frac{1 - p}{p^{2}}$ .

Step by step solution

Given Information

The variance of a geometric random variable $X$ having parameter $p$ .

Explanation

Suppose a sequence of independent Bernoulli random variables ${(X_{n})}_{n \geq 1}$ with the parameter of success $p$ . Let $X$ mark the index of first random variable which has yielded a success. We know that $Z$ has Geometric distribution with parameter $p$ . Using the formula for conditional variance, we can condition on random variable $X_{1}$ . We have that

Var (X) = E (Var (X ∣ X_{1})) + Var (E (X ∣ X_{1}))

$E (Var (X ∣ X_{1})) = E (E (X^{2} ∣ X_{1}) - E {(X ∣ X_{1})}^{2})$

$= P (X_{1} = 0) \cdot (E (X^{2} ∣ X_{1} = 0) - E {(X ∣ X_{1} = 0)}^{2})$ + $P (X_{1} = 1) (E (X^{2} ∣ X_{1} = 1) - E {(X ∣ X_{1} = 1)}^{2})$

$= (1 - p) (E (X^{2}) - E (X)^{2}) + 0$

$= (1 - p) Var (X)$

Explanation

$E (X ∣ X_{1}) = 1 \cdot I_{(X_{1} = 1)} + (1 + \frac{1}{p}) I_{(X_{1} = 0)}$

Applying the variance,

$Var (E (X ∣ X_{1})) = p (1 - p) + {(1 + \frac{1}{p})}^{2} p (1 - p) - 2 p (1 - p) (1 + \frac{1}{p})$

$Var (X) = (1 - p) Var (X) + {(\frac{1}{p})}^{2} p (1 - p)$

$Var (X) = \frac{1 - p}{p^{2}}$

Final Answer

The variance of a geometric random variable $X$ having parameter $p$ is $Var (X) = \frac{1 - p}{p^{2}}$ .

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Use the conditional variance formula to determine the variance of a geometric random variable $X$ having parameter $p$ .

Short Answer

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Use the conditional variance formula to determine the variance of a geometric random variable X having parameter p.

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Use the conditional variance formula to determine the variance of a geometric random variable $X$ having parameter $p$ .