/*! 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 10 Let \(z^{*}\) be drawn at random... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 10

Let \(z^{*}\) be drawn at random from the discrete distribution which has mass \(n^{-1}\) at each point \(z_{i}=x_{i}-\bar{x}+\mu_{0}\), where \(\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is the realization of a random sanple. Determine \(E\left(z^{*}\right)\) and \(V\left(z^{*}\right)\).

Short Answer

Expert verified
The expected value of \(z^{*}\) is \(\mu_{0}\) and the variance of \(z^{*}\) is \(\sigma^{2}\).

Step by step solution

01

Define \(z^{*}\) and calculate its expected value

Starting with each point \(z_{i}=x_{i}-\bar{x}+\mu_{0}\), let's calculate the expected value \(E(z^{*})\). Since all \(x_{i}\) have uniform probability of \(1/n\), the expected value becomes \(E\left(z^{*}\right)=\frac{1}{n}\sum_{i=1}^{n}z_{i}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x}+\mu_{0})\), which simplifies to \(\mu_{0}\) with the properties of mean and the definitions of \(x_i\) and \(\bar{x}\).
02

Calculate the variance of \(z^{*}\)

Now, let's calculate the variance \(V(z^{*})\). Following the definition of variance, we should compute \(V\left(z^{*}\right)=E\left[{(z^{*}-E(z^{*}))}^{2}\right]\). Substituting \(z^{*}\) and \(E(z^{*})\) into the formula, we get \(V\left(z^{*}\right)=E\left[{(x_i-\bar{x})}^{2}\right]=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^{2}\), which is the variance of the \(x_i\)'s, denoted as \(\sigma^{2}\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value
The expected value is a fundamental concept in probability, often denoted as E, which provides a measure of the center or the average outcome one would anticipate from a probability distribution. Imagine you're playing a game that has various outcomes, each with a specific payoff. The expected value will tell you, on average, how much you might win or lose per play in the long run.

To calculate the expected value of a discrete random variable, such as the one represented by z* in our exercise, we sum up the products of each possible value the variable can take and its probability of occurring. In mathematical terms, it is expressed as E(z*) = 鈭�(value 脳 probability). For our specific problem, since z* have equal chances of being any value zi, the expected value turns out to be the given constant 渭0.
Variance
Variance measures the spread of a set of numbers, sometimes referred to as their
Random Sample
You may have heard the term random sample tossed around in statistics, but what does it really mean? It's quite simple: a random sample is a subset of individuals chosen from a larger set, or population, where each individual has an equal chance of being selected. Consider it a mini-representation of the entire group.

In our exercise, the set (x1, x2, ..., xn) represents a random sample from a larger population. The random sample serves a crucial role in inferential statistics, where estimates or predictions about the larger population are made based on the data collected from the sample.

However, it is vital that the random sample is just that鈥攔andom. This ensures that it is unbiased and representative of the whole population, allowing for accurate inferences. Our analysis of the expected value and variance relies on this randomness to provide meaningful insights into the population's characteristics based on the sample.

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 \(X_{1}, X_{2}, \ldots, X_{n}\) be random sample from a distribution of either type. A measure of spread is Gini's mean difference $$ G=\sum_{j=2}^{n} \sum_{i=1}^{j-1}\left|X_{i}-X_{j}\right| /\left(\begin{array}{l} n \\ 2 \end{array}\right) $$ (a) If \(n=10\), find \(a_{1}, a_{2}, \ldots, a_{10}\) so that \(G=\sum_{i=1}^{10} a_{i} Y_{i}\), where \(Y_{1}, Y_{2}, \ldots, Y_{10}\) are the order statistics of the sample. (b) Show that \(E(G)=2 \sigma / \sqrt{\pi}\) if the sample arises from the normal distribution \(N\left(\mu, \sigma^{2}\right)\)

. Similar to Exercise \(5.8 .1\) but now approximate \(\int_{0}^{1.96} \frac{1}{\sqrt{2 \pi}} \exp \left\\{-\frac{1}{2} t^{2}\right\\} d t\).

Using the assumptions behind the confidence interval given in expression (5.4.17), show that $$ \sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}} / \sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}} \stackrel{P}{\rightarrow} 1 $$

. For \(\alpha>0\) and \(\beta>0\), consider the following accept/reject algorithm: (1) Generate \(U_{1}\) and \(U_{2}\) iid uniform \((0,1)\) random variables. Set \(V_{1}=U_{1}^{1 / \mathrm{a}}\) and \(V_{2}=U_{2}^{1 / \beta}\) (2) Set \(W=V_{1}+V_{2}\). If \(W \leq 1\), set \(X=V_{1} / W\), else go to Step (1). (3) Deliver \(X\).

Let \(X_{1}, X_{2}, \ldots, X_{n}, X_{n+1}\) be a random sample of size \(n+1, n>1\), from a distribution that is \(N\left(\mu, \sigma^{2}\right) .\) Let \(\bar{X}=\sum_{1}^{n} X_{i} / n\) and \(S^{2}=\sum_{1}^{n}\left(X_{i}-\bar{X}\right)^{2} /(n-1)\). Find the constant \(c\) so that the statistic \(c\left(\bar{X}-X_{n+1}\right) / S\) has a \(t\) -distribution. If \(n=8\), determine \(k\) such that \(P\left(\bar{X}-k S
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.