Numerade

A First Course in Probability

Sheldon Ross

ISBN #9780136033134

8th Edition

502 Questions

32,063 Students Helped

Summary

This section introduces continuous random variables and emphasizes the role of the probability density function in calculating probabilities through integration. Key topics include the computation of expectation and variance, and the analysis of several important continuous distributions such as uniform, normal, exponential, gamma, Weibull, Cauchy, and beta. The chapter also explains how to transform random variables and introduces the concept of the hazard rate, which is essential in evaluating the lifetime of systems in reliability engineering.

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 2

A system consisting of one original unit plus a spare can function for a random amount of time $X$ If the density of $X$ is given (in units of months) by $$f(x)=\left\{\begin{array}{ll}C x e^{-x / 2} & x>0 \\0 & x \leq 0\end{array}\right.$$ what is the probability that the system functions for at least 5 months?

Example 3

Consider the function $$f(x)=\left\{\begin{array}{ll}C\left(2 x-x^{3}\right) & 0<x<\frac{5}{2} \\0 & \text { otherwise }\end{array}\right.$$ Could $f$ be a probability density function? If so, determine $C .$ Repeat if $f(x)$ were given by $$f(x)=\left\{\begin{array}{ll}C\left(2 x-x^{2}\right) & 0<x<\frac{5}{2} \\0 & \text { otherwise }\end{array}\right.$$

Example 4

The probability density function of $X$, the lifetime of a certain type of electronic device (measured in hours), is given by $$ f(x)=\left\{\begin{array}{ll}\frac{10}{x^{2}} & x>10 \\0 & x \leq 10\end{array}\right.$$ (a) Find $P\{X>20\}$ (b) What is the cumulative distribution function of $X ?$ (c) What is the probability that, of 6 such types of devices, at least 3 will function for at least 15 hours? What assumptions are you making?

Example 5

A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function $$f(x)=\left\{\begin{array}{ll}5(1-x)^{4} & 0<x<1 \\0 & \text { otherwise }\end{array}\right.$$ what must the capacity of the tank be so that the probability of the supply's being exhausted in a given week is .01?