Chapter 8: Q. 8.24 (page 392)

It $X$ is a Poisson random variable with a mean $100$ , then $P X > 120$ is approximately
$(a) . 02,$
$(b) . 5$ or
$(c) . 3 ?$

Short Answer

Expert verified

Then, $P X > 120$ is approximately $(a) . 02 .$

Step by step solution

Given Information.

$X$ is a Poisson random variable with a mean $100$ .

Explanation.

Suppose that $X_{1}, \dots, X_{100}$ is independent and equally distributed with the distribution $Pois (1)$ . Therefore

$X = \sum_{i} X_{i} ~ Pois (100)$

Using the Central Limit Theorem, we can approximate that

$X ~ A N (n E (X_{1}), Var (X_{1})) = A N (100, 1)$

Therefore, we have an approximation

$P (X > 120) = P (\frac{X - n E (X_{1})}{\sqrt{Var (X_{1})}} > \frac{120 - n E (X_{1})}{\sqrt{Var (X_{1})}})$

$= P (\frac{X - n E (X_{1})}{\sqrt{Var (X_{1})}} > 20) \approx 1 - Φ (20) = 0.02$

so the right answer is $(a)$ .

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It $X$ is a Poisson random variable with a mean $100$ , then $P X > 120$ is approximately
$(a) . 02,$
$(b) . 5$ or
$(c) . 3 ?$

Short Answer

Step by step solution

Given Information.

Explanation.

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ItXis a Poisson random variable with a mean100, thenPX>120is approximately(a).02,(b).5or(c).3?

Short Answer

Step by step solution

Given Information.

Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

It $X$ is a Poisson random variable with a mean $100$ , then $P X > 120$ is approximately
$(a) . 02,$
$(b) . 5$ or
$(c) . 3 ?$