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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 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

01

Given information:

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

02

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" $F_{Y} (y) = P (Y \leq y) = P (e^{X} \leq y) = P (X \leq \ln (y)) = F_{X} (\ln (y))$ , which is the CDF with respect to $X$ .

03

Explanation:

After taking the derivative to get the PDF we obtain $f_{Y} (y) = \frac{d}{d y} F_{X} (\ln (y)) = F_{X} (\ln (y)) \times \frac{1}{y}$ 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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Most popular questions from this chapter

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

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

$(b)$ the density function of the random variable $|X|$ .

Define a collection of events $E_{a}, 0 < a < 1$ having the property that $P (E_{a}) = 1$ for all $a$ but $P (\underset{a}{\cap} E_{a}) = 0$

Hint: Let $X$ be uniform over $(0, 1)$ and define each $E_{a}$ in terms of $X$ .

Show that if $X$ has density function $f$ , then $E [g (X)] = \int_{- \infty}^{\infty} g (x) f (x) d x$

Hint: Using Theoretical Exercise $5.2$ , start with $E [g (X)] = \int_{0}^{\infty} P {g (X) > y} d y - \int_{0}^{\infty} P {g (X) < - y} d y$ and then proceed as in the proof given in the text when $g (X) \geq 0 .$

The number of years a radio function is exponentially distributed with the parameter $λ = \frac{1}{8}$ If Jones buys a used radio, what is the probability that it will be working after an additional $8$ year?

Consider the beta distribution with parameters $(a, b)$ . Show that

(a) when $a > 1$ and $b > 1$ , the density is unimodal (that is, it has a unique mode) with mode equal to $(a - 1) / (a + b - 2)$

(b) when $a \leq 1$ , $b \leq 1$ , and $a + b < 2$ , the density is either unimodal with mode at $0$ or $1$ or U-shaped with modes at both $0$ and $1$ ;

(c) when $a = 1 = b$ , all points in $[0, 1]$ are modes.

See all solutions

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