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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. 5.18 (page 216)

Verify that the gamma density function integrates to $1$ .

Short Answer

Expert verified

To establish the assertion, integrate $F$ over $鈩�$ and analyze the definition of the Gamma function.

Step by step solution

01

Find The density function.

Suppose that $X ~$ Gamma $(伪, 位)$ The density function is

$f (x) = \frac{位^{伪}}{螕 (伪)} x^{伪 - 1} e^{- 位 x}$

For $x 鈮� 0$ otherwise it is equal to zero. We want to prove that

${鈭�}_{鈩�} f (x) d x = 1$

We have that,

localid="1649661055597" ${鈭�}_{鈩�} f (x) d x = {鈭�}_{0}^{鈭�} \frac{位^{伪}}{螕 (伪)} x^{伪 - 1} e^{- 位 x} d x$

= \frac{位^{伪}}{螕 (伪)} {鈭�}_{0}^{鈭�} x^{伪 - 1} e^{- 位 x} d x

(1)

02

Evaluate the integral process.

In order to evaluate the integral, make the substitution $s = 位 x$ that yields $x = \frac{s}{位}$ and $d s = 位 d x$ we have that

localid="1649661029116" ${鈭�}_{0}^{鈭�} x^{伪 - 1} e^{- 位 x} d x = {鈭�}_{0}^{鈭�} {(\frac{s}{位})}^{伪 - 1} e^{- s} \frac{d s}{位}$

$= \frac{1}{位^{伪}} {鈭�}_{0}^{鈭�} s^{伪 - 1} e^{- s} d s = \frac{螕 (伪)}{位^{伪}}$

If that plug $(1)$ , we end up with

localid="1649661004300" $\frac{位^{伪}}{螕 (伪)} {鈭�}_{0}^{鈭�} x^{伪 - 1} e^{- 位 x} d x$

$= \frac{位^{伪}}{螕 (伪)} 脳 \frac{螕 (伪)}{位^{伪}} = 1$

Hence, we have proved the claim.

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Most popular questions from this chapter

Let $X$ be a random variable with probability density function

$f (x) = \{\begin{cases} c (1 - x^{2}) 鈭� 1 < x < 1 \\ 0 otherwise \end{cases}$

(a) What is the value of $c$ ?

(b) What is the cumulative distribution function of $X$ ?

For any real number $y$ , define $y^{+}$ by

$y^{+} = \begin{array}{l} y, & if y 鈮� 0 \\ 0, & if y < 0 \end{array}$

Let $c$ be a constant.

(a) Show that

$E [(Z - c)^{+}] = \frac{1}{\sqrt{2 蟺}} e^{- c^{2} / 2} - c (1 - 桅 (c))$

when $Z$ is a standard normal random variable.

(b) Find $E [(X - c)^{+}]$ when $X$ is normal with mean $渭$ and variance $蟽^{2}$ .

With $桅 (x)$ being the probability that a normal random variable with mean $0$ and variance $1$ is less than $x$ , which of the following are true:

(a) $桅 (- x) = 桅 (x)$

(b) $桅 (x) + 桅 (- x) = 1$

(c) $桅 (- x) = 1 / 桅 (x)$

Find the distribution of $R = A \sin (胃)$ , where $A$ is a fixed constant and $胃$ is uniformly distributed on $(\frac{- 蟺}{2}, \frac{蟺}{2})$ . Such a random variable $R$ arises in the theory of ballistics. If a projectile is fired from the origin at an angle $伪$ from the earth with a speed $谓$ , then the point $R$ at which it returns to the earth can be expressed as $R = (\frac{v^{2}}{g}) \sin (2 伪)$ , where $g$ is the gravitational constant, equal to $980$ centimeters per second squared.

Compute the hazard rate function of a gamma random variable with parameters $(伪, 位)$ and show it is increasing when $伪$ $鈮� 1$ and decreasing when $伪鈮� 1$

See all solutions

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