(a) If $ X_1, X_2, \ldots, X_8 \stackrel{\text{i.i.d.}}{\sim} N(0, 52) $, then the distribution of $\frac{\bar{X}}{\sqrt{52/8}}$ is (A) $ N\left(0, \frac{\bar{X}}{\sqrt{52/8}}\right) $;(B) $ N\left(0, \frac{1}{\sqrt{52/8}}\right) $;(C) $ N(0, 1) $;(D) $ N(\bar{x}, 1) $;(E) $ \text{Unif}[0, 1] $.(b) If $ B_1 \sim \text{Binom}(1, p) $ and $ B_2 \sim \text{Bernoulli}(p) $, independently of each other, then $ B_1 + B_2 $ follows(A) $\text{Binom}(1, p)$;(B) $\text{Bernoulli}(p)$;(C) $\text{Binom}(2, p)$;(D) $\text{Binom}(3, p^2)$;(E) none of these.

Answered: (a) If X1,X2, ..,Xg N(0, 52), then the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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(a) If $ X_1, X_2, \ldots, X_8 \stackrel{\text{i.i.d.}}{\sim} N(0, 52) $, then the distribution of $\frac{\bar{X}}{\sqrt{52/8}}$ is

(A) $ N\left(0, \frac{\bar{X}}{\sqrt{52/8}}\right) $;

(B) $ N\left(0, \frac{1}{\sqrt{52/8}}\right) $;

(C) $ N(0, 1) $;

(D) $ N(\bar{x}, 1) $;

(E) $ \text{Unif}[0, 1] $.

(b) If $ B_1 \sim \text{Binom}(1, p) $ and $ B_2 \sim \text{Bernoulli}(p) $, independently of each other, then $ B_1 + B_2 $ follows

(A) $\text{Binom}(1, p)$;

(B) $\text{Bernoulli}(p)$;

(C) $\text{Binom}(2, p)$;

(D) $\text{Binom}(3, p^2)$;

(E) none of these.

Expert Solution

Step 1

(a)

Given information,

$X_{i}' s$ are iids, that is, independent and identically distributed.

$X_{1}, X_{2}, . . . . ., X_{8} ~ N (0, 52)$

Answer and explanation:

From the Central Limit Theorem, if $X_{1}, . . . . . . ., X_{n}$ are independent and identically distributed random variables with mean $μ$ and variance $σ^{2}$ , then:

$\bar{X} ~ N (μ, \frac{σ^{2}}{n}) \Rightarrow \frac{\bar{X} - μ}{\sqrt{σ^{2} / n}} ~ N (0, 1)$

Since, the value of $μ$ is 0 in this case, therefore, $\frac{\bar{X}}{\sqrt{52 / 8}} ~ N (0, 1)$ .

Hence, the correct answer is option (C), that is, $\frac{\bar{X}}{\sqrt{52 / 8}} ~ N (0, 1)$ .

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