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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 7: Q.7.7I (page 358)

In Problem 7.70, suppose that the coin is tossed $n$ times. Let $X$ denote the number of heads that occur. Show that $P {X = i} = \frac{1}{n + 1} i = 0, 1, \dots, n$
Hint: Make use of the fact that,
$\int_{0}^{1} x^{a - 1} (1 - x)^{b - 1} d x = \frac{(a - 1)! (b - 1)!}{(a + b - 1)!}$
When $a$ and $b$ are positive integers.

Short Answer

Expert verified

It has been shown that $P {X = i} = \frac{1}{n + 1}; i = 0, 1, \dots, n$

Step by step solution

01

Given Information

Number of times coins tossed $= n$

Number of heads occur $= X$

Use: $\int_{0}^{1} x^{a - 1} (1 - x)^{b - 1} d x = \frac{(a - 1)! (b - 1)!}{(a + b - 1)!}$

Positive integers $= a, b$

02

Explanation

$X =$ Number of heads that occur

If coin is tossed $n$ times then $X$ can take value $0.1.2 \dots \dots, n$

$P [H e a d s o c c u r] = p ~ U (0, 1)$

$P [X = i] = \int_{0}^{1} (\begin{array}{l} n \\ i \end{array}) p_{i} (1 - p)^{n - i} dp; i = 0, 1, 2, \dots, n$

$= \frac{n!}{(i)! (n - i)!} \int_{0}^{1} p^{i} (1 - p)^{n - i} dp$

$= \frac{n!}{(i)! (n - i)!} \frac{(i)! (n - i)!}{(n + 1)!}$

Since

$\int_{0}^{1} x^{a - 1} (1 - x)^{b - 1} d x = \frac{(a - 1)! (b - 1)!}{(a + b - 1)!}$

$∴ P [X = i] = \frac{1}{n + 1}; i = 0, 1, 2, \dots, n$

03

Final Answer

Hence, it has been shown that $P {X = i} = \frac{1}{n + 1}; i = 0, 1, \dots, n$

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Most popular questions from this chapter

Let $X_{1}, X_{2}, \dots$ be a sequence of independent random variables having the probability mass function $P \{X_{n} = 0\} = P \{X_{n} = 2\} = 1 / 2, n \geq 1$

The random variable $X = \sum_{n = 1}^{\infty} X_{n} / 3^{n}$ is said to have the Cantor distribution.

Find $E [X]$ and $V a r (X) .$

In Problem 7.9, compute the variance of the number of empty urns.

A certain region is inhabited by r distinct types of a certain species of insect. Each insect caught will, independently of the types of the previous catches, be of type i with probability

$P_{i}, i = 1, \dots, r \sum_{1}^{r} P_{i} = 1$

(a) Compute the mean number of insects that are caught before the ﬁrst type $1$ catch.

(b) Compute the mean number of types of insects that are caught before the ﬁrst type $1$ catch.

The joint density of $X$ and $Y$ is given by

$f (x, y) = \frac{e^{- y}}{y}, 0 < x < y, 0 < y < \infty$

Compute $E [X^{3} ∣ Y = y]$ .

Consider the following dice game: A pair of dice is rolled. If the sum is $7,$ then the game ends and you win $0 .$ If the sum is not $7,$ then you have the option of either stopping the game and receiving an amount equal to that sum or starting over again. For each value of $i, i = 2, . . ., 12,$ find your expected return if you employ the strategy of stopping the first time that a value at least as large as $i$ appears. What value of $i$ leads to the largest expected return? Hint: Let $X i$ denote the return when you use the critical value $i .$ To compute $E [X i]$ , condition on the initial sum.

See all solutions

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