Chapter 8: Q 8.4 (page 390)

Let X₁, ... , X₂₀ be independent Poisson random variables with mean 1.
(a) Use the Markov inequality to obtain a bound on
$P \{{\sum X_{i} > 15}_{1}^{20}\}$
(b) Use the central limit theorem to approximate
$P \{{\sum X_{i} > 15}_{1}^{20}\}$

Short Answer

Expert verified

a) $P {{\sum X_{i}}_{i = 1}^{20} \geq 15} \leq \frac{4}{3}$

b) $P {{\sum X_{i}}_{i = 1}^{20} \geq 15} = 0.86$

Step by step solution

Step 1. Given information

X₁, ... , X₂₀ be independent Poisson random variables.

Mean = $λ = 1$

Step 2. a) By Markov's inequality

$P {{\sum X_{i}}_{i = 1}^{20} \geq 15} \leq \frac{E (\underset{i}{\sum X_{i}})}{15} = \frac{20}{15} = \frac{4}{3}$

Step 3. b) By central limit theorem

$P {{\sum X_{i}}_{i = 1}^{20} \geq 15} = P {\underset{i}{\sum X_{i}} - 20 \geq 15 - 20} P {{\sum X_{i}}_{i = 1}^{20} \geq 15} = P {\underset{i}{\sum X_{i}} - 20 \geq - 5}$

Step 4. Simplification

$P {{\sum X_{i}}_{i = 1}^{20} \geq 15} = P {\frac{\bar{X} - 1}{\frac{1}{\sqrt{20}}} \geq - . 25 \sqrt{20}} P {{\sum X_{i}}_{i = 1}^{20} \geq 15} = P {Z \geq - 1.18} = 0.86$

Step 5. Final answer

a) $P {{\sum X_{i}}_{i = 1}^{20} \geq 15} \leq \frac{4}{3}$

b) $P {{\sum X_{i}}_{i = 1}^{20} \geq 15} = 0.86$

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Let X₁, ... , X₂₀ be independent Poisson random variables with mean 1.
(a) Use the Markov inequality to obtain a bound on
$P \{{\sum X_{i} > 15}_{1}^{20}\}$
(b) Use the central limit theorem to approximate
$P \{{\sum X_{i} > 15}_{1}^{20}\}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. a) By Markov's inequality

Step 3. b) By central limit theorem

Step 4. Simplification

Step 5. Final answer

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Let X1, ... , X20 be independent Poisson random variables with mean 1.(a) Use the Markov inequality to obtain a bound onP∑Xi>15120(b) Use the central limit theorem to approximateP∑Xi>15120

Short Answer

Step by step solution

Step 1. Given information

Step 2. a) By Markov's inequality

Step 3. b) By central limit theorem

Step 4. Simplification

Step 5. Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Discrete Mathematics

Mechanics Maths

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Let X₁, ... , X₂₀ be independent Poisson random variables with mean 1.
(a) Use the Markov inequality to obtain a bound on
$P \{{\sum X_{i} > 15}_{1}^{20}\}$
(b) Use the central limit theorem to approximate
$P \{{\sum X_{i} > 15}_{1}^{20}\}$