/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Let \(\bar{X}\) denote the mean ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 2

Let \(\bar{X}\) denote the mean of a random sample of size 128 from a gamma distribution with \(\alpha=2\) and \(\beta=4\). Approximate \(P(7<\bar{X}<9)\).

Short Answer

Expert verified
The process involves identifying parameters of the gamma distribution, applying the Central Limit Theorem to standardize the limits, and finally finding the probability using the standard normal distribution. The actual value of the probability would depend on finding the area under the normal curve between the standardized limits \(Z_1\) and \(Z_2\), which can be achieved through the use of statistical tables or software.

Step by step solution

01

Identify parameters

Here, shape parameter \(\alpha=2\) and scale parameter \(\beta=4\). The mean and variance of a gamma distribution are given respectively by \(\alpha\beta\) and \(\alpha\beta^2\). So, \(\mu = \alpha\beta = 2*4 = 8\) and \(\sigma^2 = \alpha\beta^2 = 2*(4^2) = 32\). The standard deviation will be \(\sigma = \sqrt{32}\).
02

Apply Central Limit Theorem

By the Central Limit Theorem, the distribution of sample means will be approximately normally distributed for large sample sizes. Thus, standardized value \(Z\) for the random variable \(X\) can be calculated using the formula \(Z = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\) where \(n\) is the sample size, \(\mu\) is the population mean, and \(\sigma\) is the standard deviation of the population.
03

Standardize the limits

We need to standardize 7 and 9 using the formula from Step 2. So, \(Z_1 = \frac{(7-8)}{\sqrt{32/128}}\) and \(Z_2 = \frac{(9-8)}{\sqrt{32/128}}\). Now, we're looking to find the probability \(P(Z_1<Z<Z_2)\).
04

Find the probability

This can be found using the standard normal distribution table or software that can compute statistical functions. The probability \(P(7<\bar{X}<9)\) is equivalent to the probability \(P(Z_1<Z<Z_2)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(\left\\{a_{n}\right\\}\) be a sequence of real numbers. Hence, we can also say that \(\left\\{a_{n}\right\\}\) is a sequence of constant (degenerate) random variables. Let \(a\) be a real number. Show that \(a_{n} \rightarrow a\) is equivalent to \(a_{n} \stackrel{P}{\rightarrow} a\)

Let \(Y_{n}\) denote the maximum of a random sample of size \(n\) from a distribution of the continuous type that has cdf \(F(x)\) and pdf \(f(x)=F^{\prime}(x) .\) Find the limiting distribution of \(Z_{n}=n\left[1-F\left(Y_{n}\right)\right]\).

Let \(X_{1}, \ldots, X_{n}\) be iid random variables with common pdf $$f(x)=\left\\{\begin{array}{ll} e^{-(x-\theta)} & x>\theta-\infty<\theta<\infty \\ 0 & \text { elsewhere }\end{array}\right.$$ This pdf is called the shifted exponential. Let \(Y_{n}=\min \left\\{X_{1}, \ldots, X_{n}\right\\} .\) Prove that \(Y_{n} \rightarrow \theta\) in probability, by first obtaining the cdf of \(Y_{n}\).

Let \(Y_{2}\) denote the second smallest item of a random sample of size \(n\) from a distribution of the continuous type that has cdf \(F(x)\) and pdf \(f(x)=F^{\prime}(x)\). Find the limiting distribution of \(W_{n}=n F\left(Y_{2}\right)\).

Let \(Y_{1}
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.