A First Course in Probability (10th Edition)

10th Edition

ISBN: 9780134753119

Author: Sheldon Ross

Publisher: PEARSON

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Textbook Question

Chapter 5, Problem 5.21STPE

With Φ ( x ) being the probability that a normal random variable with mean 0 and variance 1 is less than x. which of the following are true:

a. Φ ( − x ) = Φ ( x )

b. Φ ( x ) + Φ ( − x ) = 1

c. Φ ( − x ) = 1 Φ ( x )

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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Chapter 5, Problem 5.20STPE

Chapter 5, Problem 5.22STPE

Chapter 5 Solutions

A First Course in Probability (10th Edition)

Ch. 5 - Let X be a random variable with probability...Ch. 5 - Prob. 5.2P Ch. 5 - Prob. 5.3P Ch. 5 - The probability density function of X. the...Ch. 5 - Prob. 5.5P Ch. 5 - Compute E[X] if X has a density function given by...Ch. 5 - The density function of X is given by...Ch. 5 - The lifetime in hours of an electronic tube is a...Ch. 5 - Consider Example 4b &I of Chapter 4 &I, but now...Ch. 5 - Trains headed for destination A arrive at the...

Ch. 5 - A point is chosen at random on a line segment of...Ch. 5 - A bus travels between the two cities A and B....Ch. 5 - You arrive at a bus stop at 10A.M., knowing that...Ch. 5 - Let X be a uniform (0, 1) random variable. Compute...Ch. 5 - If X is a normal random variable with parameters...Ch. 5 - The annual rainfall (in inches) in a certain...Ch. 5 - The salaries of physicians in a certain speciality...Ch. 5 - Suppose that X is a normal random variable with...Ch. 5 - Let be a normal random variable with mean 12 and...Ch. 5 - If 65 percent of the population of a large...Ch. 5 - Suppose that the height, in inches, of a...Ch. 5 - Every day Jo practices her tennis serve by...Ch. 5 - One thousand independent rolls of a fair die will...Ch. 5 - The lifetimes of interactive computer chips...Ch. 5 - Each item produced by a certain manufacturer is,...Ch. 5 - Two types of coins are produced at a factory: a...Ch. 5 - In 10,000 independent tosses of a coin, the coin...Ch. 5 - Twelve percent of the population is left handed....Ch. 5 - A model for the movement of a stock supposes that...Ch. 5 - An image is partitioned into two regions, one...Ch. 5 - a. A fire station is to be located along a road of...Ch. 5 - The time (in hours) required to repair a machine...Ch. 5 - If U is uniformly distributed on (0,1), find the...Ch. 5 - Jones figures that the total number of thousands...Ch. 5 - Prob. 5.35P Ch. 5 - The lung cancer hazard rate (t) of a t-year-old...Ch. 5 - Suppose that the life distribution of an item has...Ch. 5 - If X is uniformly distributed over (1,1), find (a)...Ch. 5 - Prob. 5.39P Ch. 5 - If X is an exponential random variable with...Ch. 5 - If X is uniformly distributed over(a,b), find a...Ch. 5 - Prob. 5.42P Ch. 5 - Find the distribution of R=Asin, where A is a...Ch. 5 - Let Y be a log normal random variable (see Example...Ch. 5 - The speed of a molecule in a uniform gas at...Ch. 5 - Show that E[Y]=0P{Yy}dy0P{Yy}dy Hint: Show that...Ch. 5 - Show that if X has density function f. then...Ch. 5 - Prob. 5.4TE Ch. 5 - Use the result that for a nonnegative random...Ch. 5 - Prob. 5.6TE Ch. 5 - The standard deviation of X. denoted SD(X), is...Ch. 5 - Let X be a random variable that takes on values...Ch. 5 - Show that Z is a standard normal random variable;...Ch. 5 - Let f(x) denote the probability density function...Ch. 5 - Let Z be a standard normal random variable Z and...Ch. 5 - Use the identity of Theoretical Exercises 5.5 .Ch. 5 - The median of a continuous random variable having...Ch. 5 - The mode of a continuous random variable having...Ch. 5 - If X is an exponential random variable with...Ch. 5 - Compute the hazard rate function of X when X is...Ch. 5 - If X has hazard rate function X(t), compute the...Ch. 5 - Prob. 5.18TE Ch. 5 - If X is an exponential random variable with mean...Ch. 5 - Prob. 5.20TE Ch. 5 - Prob. 5.21TE Ch. 5 - Compute the hazard rate function of a gamma random...Ch. 5 - Compute the hazard rate function of a Weibull...Ch. 5 - Prob. 5.24TE Ch. 5 - Let Y=(Xv) Show that if X is a Weibull random...Ch. 5 - Let F be a continuous distribution function. If U...Ch. 5 - If X is uniformly distributed over (a,b), what...Ch. 5 - Consider the beta distribution with parameters...Ch. 5 - Prob. 5.29TE Ch. 5 - Prob. 5.30TE Ch. 5 - Prob. 5.31TE Ch. 5 - Let X and Y be independent random variables that...Ch. 5 - Prob. 5.33TE Ch. 5 - The number of minutes of playing time of a certain...Ch. 5 - For some constant c. the random variable X has the...Ch. 5 - Prob. 5.3STPE Ch. 5 - Prob. 5.4STPE Ch. 5 - The random variable X is said to be a discrete...Ch. 5 - Prob. 5.6STPE Ch. 5 - To be a winner in a certain game, you must be...Ch. 5 - A randomly chosen IQ test taker obtains a score...Ch. 5 - Suppose that the travel time from your home to...Ch. 5 - The life of a certain type of automobile tire is...Ch. 5 - The annual rainfall in Cleveland, Ohio, is...Ch. 5 - Prob. 5.12STPE Ch. 5 - Prob. 5.13STPE Ch. 5 - Prob. 5.14STPE Ch. 5 - The number of years that a washing machine...Ch. 5 - Prob. 5.16STPE Ch. 5 - Prob. 5.17STPE Ch. 5 - There are two types of batteries in a bin. When in...Ch. 5 - Prob. 5.19STPE Ch. 5 - For any real number y define byy+=y,ify00,ify0 Let...Ch. 5 - With (x) being the probability that a normal...Ch. 5 - Prob. 5.22STPE Ch. 5 - Letf(x)={13ex1313e(x1)ifx0if0x1ifx1 a. Show that f...Ch. 5 - Prob. 5.24STPE

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With Φ ( x ) being the probability that a normal random variable with mean 0 and variance 1 is less than x. which of the following are true: a. Φ ( − x ) = Φ ( x ) b. Φ ( x ) + Φ ( − x ) = 1 c. Φ ( − x ) = 1 Φ ( x )

Solution Summary: The author explains that the correct option is "b". The normal distribution is a symmetrical distribution.

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Chapter 5 Solutions