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Mathematical Statistics and Data Analysis

John A. Rice

Chapter 6

Distributions Derived from the Normal Distribution - all with Video Answers

Educators

Chapter Questions

01:43

Problem 1

Prove Proposition A of Section 6.2.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:43

Problem 2

Prove Proposition B of Section 6.2.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:13

Problem 3

Let $\bar{X}$ be the average of a sample of 16 independent normal random variables with mean 0 and variance $1 .$ Determine $c$ such that
$$
P(|\bar{X}|<c)=.5
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
02:46

Problem 4

If $T$ follows a $t_{7}$ distribution, find $t_{0}$ such that (a) $P\left(|T|<t_{0}\right)=.9$ and
(b) $P\left(T>t_{0}\right)=.05$

Audrey Fong
Audrey Fong
Numerade Educator
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Problem 5

Show that if $X \sim F_{n, m},$ then $X^{-1} \sim F_{m, n}$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 6

Show that if $T \sim t_{n},$ then $T^{2} \sim F_{1, n}$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:29

Problem 7

Show that the Cauchy distribution and the $t$ distribution with 1 degree of freedom are the same.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:13

Problem 8

Show that if $X$ and $Y$ are independent exponential random variables with $\lambda=1$ then $X / Y$ follows an $F$ distribution. Also, identify the degrees of freedom.

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
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Problem 9

Find the mean and variance of $S^{2},$ where $S^{2}$ is as in Section 6.3

Shu Naito
Shu Naito
Numerade Educator
03:31

Problem 10

Show how to use the chi-square distribution to calculate $P\left(a<S^{2} / \sigma^{2}<b\right)$

Raymond Matshanda
Raymond Matshanda
Numerade Educator
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Problem 11

Let $X_{1}, \ldots, X_{n}$ be a sample from an $N\left(\mu_{X}, \sigma^{2}\right)$ distribution and $Y_{1}, \ldots, Y_{m}$ be an independent sample from an $N\left(\mu_{y}, \sigma^{2}\right)$ distribution. Show how to use the
$F$ distribution to find $P\left(S_{X}^{2} / S_{Y}^{2}>c\right)$

Victor Salazar
Victor Salazar
Numerade Educator