Q 5.40 If Xis uniformly distributed ove... [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 5.40 (page 214)

If $X$ is uniformly distributed over $(0, 1)$ , find the density function of $Y = e^{X}$ .

Short Answer

Expert verified

Therefore, the PDF of $Y = e^{X} i s f_{Y} (y) = \{\begin{cases} \frac{1}{y} & i f 1 < y < e \\ 0 & o / w \end{cases}$

Step by step solution

Given information:

If $X$ is uniformly distributed over $(0, 1)$

Explanation:

First, we start with the cumulative density function $F_{Y} (y)$ , since we want to find the PDF for $Y$ . We haverole="math" localid="1646662591120" $F_{Y} (y) = P (Y 鈮� y) = P (e^{X} 鈮� y) = P (X 鈮� \ln (y)) = F_{X} (\ln (y))$ , which is the CDF with respect to $X$ .

Explanation:

After taking the derivative to get the PDF we obtain $f_{Y} (y) = \frac{d}{d y} F_{X} (\ln (y)) = F_{X} (\ln (y)) 脳 \frac{1}{y}$ by the chain rule. But since $X ~ U N I F (0, 1)$ , we know that $f_{X} (\ln (y)) = 1$ and the domain changes to $e^{0} < e^{X} < e^{1} = 1 < y < e$ , so we have that the PDF of $Y = e^{X} i s f_{Y} (y) = \{\begin{cases} \frac{1}{y} & i f 1 < y < e \\ 0 & o / w \end{cases}$

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91影视

If $X$ is uniformly distributed over $(0, 1)$ , find the density function of $Y = e^{X}$ .

Short Answer

Step by step solution

Given information:

Explanation:

Explanation:

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91影视

If Xis uniformly distributed over(0,1), find the density function ofY=eX.

Short Answer

Step by step solution

Given information:

Explanation:

Explanation:

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Most popular questions from this chapter

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Theoretical and Mathematical Physics

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If $X$ is uniformly distributed over $(0, 1)$ , find the density function of $Y = e^{X}$ .