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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 7: Q.7.45 (page 362)

Verify the formula for the moment generating function of a uniform random variable that is given in Table 7.2. Also, differentiate to verify the formulas for the mean and variance.

Short Answer

Expert verified

It has been verify the formula for the moment generating function of a uniform random variable that is given in Table as $\frac{(b 鈭� a)^{2}}{12}$ .

Step by step solution

01

Given information

The moment generating function of a uniform random variable that is given in Table

02

Solution

If $X ~ U (a, b)$

Then $M_{x} (t) = E [e^{t x}]$

$= {鈭�}_{a}^{b} 鈥� e^{t x} \frac{1}{b 鈭� a} d x$

$= \frac{1}{b 鈭� a} {[\frac{e^{t x}}{t}]}_{a}^{b}$

$= \frac{e^{b t} 鈭� e^{a t}}{t (b 鈭� a)}$

Then it has been verified

03

Solution

Now,

$M_{x} (t) = \frac{1}{(b 鈭� a)} [\frac{e^{b t} 鈭� e^{a t}}{t}]$

$= \frac{1}{(b 鈭� a)} [\frac{(1 + b t + \frac{(b t)^{2}}{2!} + \frac{(b t)^{3}}{3!} + 鈥� 鈥� .) 鈭� (1 + a t + \frac{(a t)^{2}}{2!} + \frac{(a t)^{3}}{3!} + 鈥� 鈥�)}{t}]$

$= \frac{1}{(b 鈭� a)} [\frac{(b 鈭� a) t}{t} + \frac{(b^{2} 鈭� a^{2}) t^{2}}{2! t} + \frac{(b^{3} 鈭� a^{3}) t^{3}}{3! t} + 鈥� 鈥�]$

$= 1 + \frac{(b + a)}{2} t + \frac{(b^{2} + a^{2} + a b)}{6} t^{2} + 鈥� 鈥� 鈥�$

04

Solution

Now,

$M_{x}^{鈰�} (t) = \frac{(b + a)}{2} + \frac{(b^{2} + a^{2} + a b)}{6} 2 t + 鈥� 鈥� 鈥�$

$M_{X}^{鈭�} (t) = \frac{b^{2} + a^{2} + a b}{3} + o (t); o (t) =$ Terms having higher power of t

So,

$E (X) = {M_{X}^{鈥�} (t)|}_{t = 0} = \frac{(b + a)}{2}$

$E (X^{2}) = {M_{X}^{鈭�} (t)|}_{t = 0} = \frac{(b^{2} + a^{2} + a b)}{3}$

$鈬� V (X) = E (X^{2}) 鈭� [E (X)]^{2}$

$= \frac{b^{2} + a^{2} + a b}{3} 鈭� \frac{b^{2} + a^{2} + 2 a b}{4}$

$= \frac{4 (b^{2} + a^{2} + a b) 鈭� 3 (b^{2} + a^{2} + 2 a b)}{12}$

$= \frac{(b 鈭� a)^{2}}{12}$

05

Final answer

It has been verify the formula for the moment generating function of a uniform random variable that is given in Table as $\frac{(b 鈭� a)^{2}}{12}$ .

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Most popular questions from this chapter

Consider a graph having $n$ vertices labeled $1, 2, . . ., n$ , and suppose that, between each of the $(\frac{n}{2})$ pairs of distinct vertices, an edge is independently present with probability $p$ . The degree of a vertex $i$ , designated as $D i,$ is the number of edges that have vertex $i$ as one of their vertices.

(a) What is the distribution of $D i$ ?

(b) Find $蚁 (D i, D j)$ , the correlation between $D i$ and $D j$ .

There are two misshapen coins in a box; their probabilities for landing on heads when they are flipped are, respectively, . $4$ and . $7$ . One of the coins is to be randomly chosen and flipped 10 times. Given that two of the first three flips landed on heads, what is the conditional expected number of heads in the $10$ flips?

In Problem 7.6, calculate the variance of the sum of the rolls.

Show that $Y = a + b X$ , then

$蚁 (X, Y) = \{\begin{cases} + 1 鈥呪赌呪赌呪赌� if b > 0 \\ 鈭� 1 鈥呪赌呪赌呪赌� if b < 0 \end{cases}$

A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.

(a) Let X denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find E[X].

Hint: De铿乶e n + m indicator variables, one for each of the small pills initially present and one for each of the small pills created when a large one is split in two. Now use the argument of Example $2$ m.

(b) Let Y denote the day on which the last large pills chosen. Find E[Y].

Hint: What is the relationship between X and Y?

See all solutions

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