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Fundamentals of Mathematical Statistics

S.C. Gupta, V.K. Kapoor

Chapter 2: Chapter 2

Problem 45

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Suppose that $X_{i}, i=1,2, \ldots, 20$ are independent random variables, each having a geometric distribution with parameter $0.8$. Let $S=X_{1}+\cdots+X_{20}$. Use the central limit theorem $P(X \geq 18)$.

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Step 1: Identify the Geometric Distribution

Given that $X_i, i=1,2, \ldots, 20$ are independent random variables each having a geometric distribution with parameter $0.8$. In this case, the parameter is the probability of success in each trial, denoted as $p=0.8$. The geometric distribution models the number of failures before the first success.

Step 2: Calculate Mean and Variance

For a geometrically distributed random variable $X$ with success probability $p$, the mean (expected value) $E[X]=(1-p)/p$ and the variance $Var[X]=(1-p)/p^2$. Here, mean $μ=(1-0.8)/0.8=0.25$ and variance $σ^2=(1-0.8)/(0.8)^2=0.3125$.

Step 3: Apply the Central Limit Theorem

We are asked to find $P(S \geq 18)$, where $S=X_{1}+\cdots+X_{20}$ represents the sum of 20 independent geometric random variables. We can regard $S$ as approximately normally distributed because of its large size (20). By the CLT, $S$ has mean $E[S]=n*\mu=20*0.25=5$ and variance $Var[S]=n*σ^2=20*0.3125=6.25$, thus the standard deviation $σ_S=\sqrt{Var[S]}=\sqrt{6.25}$. Then we standardize $S$ and change the inequality direction by subtracting $-18$, giving $1-P(\frac{S-5}{\sqrt{6.25}} < \frac{18-5}{\sqrt{6.25}})$

Step 4: Find the Desired Probability

We lookup the right hand side value in a standard normal distribution table or use computational software to find the associated probability. We then subtract this from one to find the required probability $P(S \geq 18)$.

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Chapter 2: Chapter 2

Suppose that \(X_{i}, i=1,2, \ldots, 20\) are independent random variables, each having a geometric distribution with parameter \(0.8\). Let \(S=X_{1}+\cdots+X_{20}\). Use the central limit theorem \(P(X \geq 18)\).

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Step 1: Identify the Geometric Distribution

Step 2: Calculate Mean and Variance

Step 3: Apply the Central Limit Theorem

Step 4: Find the Desired Probability

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