A First Course in Probability

Sheldon Ross

ISBN #9780136033134

8th Edition

502 Questions

32,063 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

Section 7.2 emphasizes the fundamental property of the expectation operator namely its linearity: the expected value of a sum equals the sum of the expected values. This principle holds true regardless of whether the underlying random variables are independent, making it a powerful tool in simplifying complex probabilistic calculations. By decomposing a random variable into indicator components, one can easily handle problems involving counts and aggregates. Additionally, understanding these properties paves the way for exploring variance, covariance, and extending concepts to more advanced areas such as conditional expectation and moment generating functions.

Learning Objectives

Key Concepts

CONCEPT

DEFINITION

Definition: Explores advanced probability models by extending fundamental theories to more complex systems. Topics include random processes (e.g., the Poisson process), Markov chains (with transition matrices, Chapman–Kolmogorov equations, and stationary distributions), and key ideas in information theory such as surprise, entropy, and coding (both noiseless and noisy).

Explores advanced probability models by extending fundamental theories to more complex systems. Topics include random processes (e.g., the Poisson process), Markov chains (with transition matrices, Chapman–Kolmogorov equations, and stationary distributions), and key ideas in information theory such as surprise, entropy, and coding (both noiseless and noisy). •

Example Problems

Example 1

A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, then one-half of the value that appears on the die. Determine her expected winnings.

Example 2

The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is randomly chosen and the object of the game is to guess the chosen three.(a) How many solutions are possible? In one version of the game, the selection is made and then each of the players is randomly given three of the remaining cards. Let $S, W$ and $R$ be, respectively, the numbers of suspects, weapons, and rooms in the set of three cards given to a specified player. Also, let $X$ denote the number of solutions that are possible after that player observes his or her three cards. (b) Express $X$ in terms of $S, W,$ and $R$ (c) Find $E[X]$

Example 3

Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let $W$ denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win. Find (a) $P(W>0\}$ (b) $P\{W<0\}$ (c) $E[W]$

Example 4

If $X$ and $Y$ have joint density function $$f_{X, Y}(x, y)=\left\{\begin{array}{ll} 1 / y, & \text { if } 0<y<1,0<x<y \\ 0, & \text { otherwise } \end{array}\right.$$ find (a) $E[X Y]$ (b) $E[X]$ (c) $E[Y]$

Example 5

The county hospital is located at the center of a square whose sides are 3 miles wide. If an accident occurs within this square, then the hospital sends out an ambulance. The road network is rectangular, so the travel distance from the hospital, whose coordinates are $(0,0),$ to the point $(x, y)$ is $|x|+|y|$ If an accident occurs at a point that is uniformly distributed in the square, find the expected travel distance of the ambulance.