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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. 5.26 (page 216)

If $x$ is a beta random variable with parameters $a$ and $b$ , show that
$E [X] = \frac{a}{a + b}$
$Var (X) = \frac{ab}{(a + b)^{2} (a + b + 1)}$

Short Answer

Expert verified

In general, find the $k$ th moment and use it to calculate the mean and variance.

Step by step solution

01

Beta distribution

Density function,

$f (x) = \frac{螕 (伪 + 尾)}{螕 (伪) 螕 (尾)} 脳 x^{伪 - 1} (1 - x)^{尾 - 1}$

$x 鈭� [0, 1]$ is to be zero.

$E (X^{k}) = {鈭�}_{鈩�} x^{k} f (x) dx = \frac{螕 (伪 + 尾)}{螕 (伪) 螕 (尾)} {鈭�}_{0}^{1} x^{k} 脳 x^{伪 - 1} (1 - x)^{尾 - 1} dx$

$= \frac{螕 (伪 + 尾)}{螕 (伪) 螕 (尾)} {鈭�}_{0}^{1} x^{k + 伪 - 1} (1 - x)^{尾 - 1} dx$

So,

localid="1649485821665" $E (X^{k}) = \frac{螕 (伪 + 尾)}{螕 (伪) 螕 (尾)} 脳 B (k + 伪, 尾)$

$= \frac{螕 (伪 + 尾)}{螕 (伪) 螕 (尾)} 脳 \frac{螕 (k + 伪) 螕 (尾)}{螕 (k + 伪 + 尾)}$

02

Explanation

For $k = 1$

$E (X) = \frac{螕 (伪 + 尾)}{螕 (伪)} 脳 \frac{螕 (伪 + 1)}{螕 (伪 + 尾 + 1)}$

$= \frac{螕 (伪 + 尾)}{螕 (伪)} 脳 \frac{伪螕 (伪)}{(伪 + 尾) 螕 (伪 + 尾)} = \frac{伪}{伪 + 尾}$

For $k = 2$

$E (X^{2}) = \frac{螕 (伪 + 尾)}{螕 (伪)} 脳 \frac{螕 (伪 + 2)}{螕 (伪 + 尾 + 2)}$

$= \frac{螕 (伪 + 尾)}{螕 (伪)} 脳 \frac{(伪 + 1) 伪螕 (伪)}{(伪 + 尾 + 1) (伪 + 尾) 螕 (伪 + 尾)}$

So,

$Var (X) = E (X^{2}) - E (X)^{2}$

$= \frac{(伪 + 1) 伪}{(伪 + 尾 + 1) (伪 + 尾)} - \frac{伪^{2}}{(伪 + 尾)^{2}}$

$= \frac{(伪 + 1) 伪 (伪 + 尾) - 伪^{2} (伪 + 尾 + 1)}{(伪 + 尾 + 1) (伪 + 尾)^{2}}$

$= \frac{伪尾}{(伪 + 尾 + 1) (伪 + 尾)^{2}}$

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Most popular questions from this chapter

Show that $E [Y] = {鈭�}_{0}^{鈭�} P \{Y > y\} d y - {鈭�}_{0}^{鈭�} P \{Y < - y\} d y$ .

Hint: Show that ${鈭�}_{0}^{鈭�} P \{Y < - y\} d y = - {鈭�}_{- 鈭�}^{0} x f_{y} (x) d x$

${鈭�}_{0}^{鈭�} P \{Y > y\} d y = {鈭�}_{0}^{鈭�} x f_{y} (x) d x$

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability $. 95$ . Approximate the probability that at most $10$ of the next $150$ items produced are unacceptable.

The mode of a continuous random variable having density $f$ is the value of $x$ for which $f (x)$ attains its maximum. Compute the mode of $x$ in cases $(a), (b)$ and $(c)$ of Theoretical Exercise $5.13$

Let X have probability density f X. Find the probability density function of the random variable Y defined by Y = a X + b.

In $10, 000$ independent tosses of a coin, the coin landed on heads $5800$ times. Is it reasonable to assume that the coin is not fair? Explain.

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