Q 5.37 聽IfX聽is uniformly distributed ... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 5: Q 5.37 (page 214)

If $X$ is uniformly distributed over $(- 1, 1),$ find
$(a) P {|X| > \frac{1}{2}}$
$(b)$ the density function of the random variable $|X|$ .

Short Answer

Expert verified

Therefore,

(a) P (|X| > \frac{1}{2}) = \frac{1}{2}

$(b)$ We have found that $Y = |X| ~ U N I F (0, 1)$

Step by step solution

Given information:

If $X$ is uniformly distributed over $(- 1, 1)$

Part (a) step 2 Explanation:

We have that $X ~ U N I F (- 1, 1)$ and wish to calculate

$P (|X| > \frac{1}{2}) = P (- \frac{1}{2} > X > \frac{1}{2}) = P (X < - \frac{1}{2} 鈭� X > \frac{1}{2}) = P (X < - \frac{1}{2}) + P (X > \frac{1}{2}) = {鈭�}_{- 1}^{\frac{1}{2}} \frac{1}{2} d x + {鈭�}_{\frac{1}{2}}^{1} \frac{1}{2} d x = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$

Part (b) step 3 Explanation:

The CDF of $Y = |X|$ , where

$F_{x} (|X|) = P (|X| 鈮� x) = P (- x 鈮� X 鈮� x) = {鈭�}_{- x}^{x} \frac{1}{2} d t = {[\frac{1}{2} t]}_{- x}^{x} = \frac{2 x}{2} = x$

for all $x 鈭� (0, 1)$ . We then differentiate the CDF to obtain the PDF as $f_{|x|} (x) = \frac{d}{d x} F (|X|) = \frac{d}{d x} (x) = 1$ for $x 鈭� (0, 1)$ . Therefore, we have found that $Y = |X| ~ U N I F (0, 1) .$

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91影视

If $X$ is uniformly distributed over $(- 1, 1),$ find
$(a) P {|X| > \frac{1}{2}}$
$(b)$ the density function of the random variable $|X|$ .

Short Answer

Step by step solution

Given information:

Part (a) step 2 Explanation:

Part (b) step 3 Explanation:

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91影视

IfXis uniformly distributed over (-1,1),find(a)P{X>12}(b)the density function of the random variableX.

Short Answer

Step by step solution

Given information:

Part (a) step 2 Explanation:

Part (b) step 3 Explanation:

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If $X$ is uniformly distributed over $(- 1, 1),$ find
$(a) P {|X| > \frac{1}{2}}$
$(b)$ the density function of the random variable $|X|$ .