10th Edition

ISBN: 9780134753676

Author: Ross

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could co…

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a…

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Textbook Question

Chapter 6, Problem 6.37TE

Suppose that (X,Y) has a bivariate normal distribution with parameters μ x , μ y , σ x , σ , ρ .

a. Show that ( X − μ x σ x , Y − μ y σ y ) has a bivariate normal distribution with parameters 0,1,0,1, ρ .

b. What is the joint distribution of ( a X + b , c Y + d ) .

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 6, Problem 6.36TE

Chapter 6, Problem 6.38TE

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Chapter 6 Solutions

EBK FIRST COURSE IN PROBABILITY, A

Ch. 6 - Two fair dice are rolled. Find the joint...Ch. 6 - Suppose that 3 balls are chosen without...Ch. 6 - In Problem 8 t, suppose that the white balls are...Ch. 6 - Repeat Problem 6.2 when the ball selected is...Ch. 6 - Repeat Problem 6.3a when the ball selected is...Ch. 6 - The severity of a certain cancer is designated by...Ch. 6 - Consider a sequence of independent Bernoulli...Ch. 6 - Prob. 6.8P Ch. 6 - The joint probability density function of X and Y...Ch. 6 - Prob. 6.10P

Ch. 6 - In Example Id, verify that f(x,y)=2exe2y,0x,0y, is...Ch. 6 - The number of people who enter a drugstore in a...Ch. 6 - A man and a woman agree to meet at a certain...Ch. 6 - An ambulance travels back and forth at a constant...Ch. 6 - The random vector (X,Y) is said to be uniformly...Ch. 6 - Suppose that n points are independently chosen at...Ch. 6 - Prob. 6.17P Ch. 6 - Let X1 and X2 be independent binomial random...Ch. 6 - Show that f(x,y)=1x, 0yx1 is a joint density...Ch. 6 - Prob. 6.20P Ch. 6 - Let f(x,y)=24xy0x1,0y1,0x+y1 and let it equal 0...Ch. 6 - The joint density function of X and Y is...Ch. 6 - Prob. 6.23P Ch. 6 - Consider independent trials, each of which results...Ch. 6 - Suppose that 106 people arrive at a service...Ch. 6 - Prob. 6.26P Ch. 6 - Prob. 6.27P Ch. 6 - The time that it takes to service a car is an...Ch. 6 - The gross daily sales at a certain restaurant are...Ch. 6 - Jills bowling scores are approximately normally...Ch. 6 - According to the U.S. National Center for Health...Ch. 6 - Monthly sales are independent normal random...Ch. 6 - Let X1 and X2 be independent normal random...Ch. 6 - Prob. 6.34P Ch. 6 - Teams 1, 2, 3, 4 are all scheduled to play each of...Ch. 6 - Let X1,...,X10 be independent with the same...Ch. 6 - The expected number of typographical errors on a...Ch. 6 - The monthly worldwide average number of airplane...Ch. 6 - In Problem 6.4, calculate the conditional...Ch. 6 - In Problem 6.3 calculate the conditional...Ch. 6 - Prob. 6.41P Ch. 6 - Prob. 6.42P Ch. 6 - Prob. 6.43P Ch. 6 - The joint probability mass function of X and Y is...Ch. 6 - Prob. 6.45P Ch. 6 - Prob. 6.46P Ch. 6 - An insurance company supposes that each person has...Ch. 6 - If X1,X2,X3 are independent random variables that...Ch. 6 - Prob. 6.49P Ch. 6 - If 3 trucks break down at points randomly...Ch. 6 - Consider a sample of size 5 from a uniform...Ch. 6 - Prob. 6.52P Ch. 6 - Let X(1),X(2),...,X(n) be the order statistics of...Ch. 6 - Let Z1 and Z2 be independent standard normal...Ch. 6 - Derive the distribution of the range of a sample...Ch. 6 - Let X and Y denote the coordinates of a point...Ch. 6 - Prob. 6.57P Ch. 6 - Prob. 6.58P Ch. 6 - Prob. 6.59P Ch. 6 - Prob. 6.60P Ch. 6 - Repeat Problem 6.60 when X and Y are independent...Ch. 6 - Prob. 6.62P Ch. 6 - Prob. 6.63P Ch. 6 - In Example 8b, let Yk+1=n+1i=1kYi. Show that...Ch. 6 - Consider an urn containing n balls numbered 1.. .....Ch. 6 - Suppose X,Y have a joint distribution function...Ch. 6 - Prob. 6.2TE Ch. 6 - Prob. 6.3TE Ch. 6 - Solve Buffons needle problem when LD.Ch. 6 - If X and Y are independent continuous positive...Ch. 6 - Prob. 6.6TE Ch. 6 - Prob. 6.7TE Ch. 6 - Let X and Y be independent continuous random...Ch. 6 - Let X1,...,Xn be independent exponential random...Ch. 6 - The lifetimes of batteries are independent...Ch. 6 - Prob. 6.11TE Ch. 6 - Show that the jointly continuous (discrete) random...Ch. 6 - In Example 5e t, we computed the conditional...Ch. 6 - Suppose that X and Y are independent geometric...Ch. 6 - Consider a sequence of independent trials, with...Ch. 6 - If X and Y are independent binomial random...Ch. 6 - Suppose that Xi,i=1,2,3 are independent Poisson...Ch. 6 - Prob. 6.18TE Ch. 6 - Let X1,X2,X3 be independent and identically...Ch. 6 - Prob. 6.20TE Ch. 6 - Suppose that W, the amount of moisture in the air...Ch. 6 - Let W be a gamma random variable with parameters...Ch. 6 - A rectangular array of mn numbers arranged in n...Ch. 6 - If X is exponential with rate , find...Ch. 6 - Suppose thatF(x) is a cumulative distribution...Ch. 6 - Show that if n people are distributed at random...Ch. 6 - Suppose that X1,...,Xn are independent exponential...Ch. 6 - Establish Equation (6.2) by differentiating...Ch. 6 - Show that the median of a sample of size 2n+1 from...Ch. 6 - Prob. 6.30TE Ch. 6 - Compute the density of the range of a sample of...Ch. 6 - Let X(1)X(2)...X(n) be the ordered values of n...Ch. 6 - Let X1,...,Xn be a set of independent and...Ch. 6 - Let X1,....Xn, be independent and identically...Ch. 6 - Prob. 6.35TE Ch. 6 - Prob. 6.36TE Ch. 6 - Suppose that (X,Y) has a bivariate normal...Ch. 6 - Suppose that X has a beta distribution with...Ch. 6 - 6.39. Consider an experiment with n possible...Ch. 6 - Prob. 6.40TE Ch. 6 - Prob. 6.41TE Ch. 6 - Each throw of an unfair die lands on each of the...Ch. 6 - The joint probability mass function of the random...Ch. 6 - Prob. 6.3STPE Ch. 6 - Let r=r1+...+rk, where all ri are positive...Ch. 6 - Suppose that X, Y, and Z are independent random...Ch. 6 - Let X and Y be continuous random variables with...Ch. 6 - The joint density function of X and Y...Ch. 6 - Consider two components and three types of shocks....Ch. 6 - Consider a directory of classified advertisements...Ch. 6 - The random parts of the algorithm in Self-Test...Ch. 6 - Prob. 6.11STPE Ch. 6 - The accompanying dartboard is a square whose sides...Ch. 6 - A model proposed for NBA basketball supposes that...Ch. 6 - Let N be a geometric random variable with...Ch. 6 - Prob. 6.15STPE Ch. 6 - You and three other people are to place bids for...Ch. 6 - Find the probability that X1,X2,...,Xn is a...Ch. 6 - 6.18. Let 4VH and Y, be independent random...Ch. 6 - Let Z1,Z2.....Zn be independent standard normal...Ch. 6 - Let X1,X2,... be a sequence of independent and...Ch. 6 - Prove the identity P{Xs,Yt}=P{Xs}+P{Yt}+P{Xs,Yt}1...Ch. 6 - In Example 1c, find P(Xr=i,Ys=j) when ji.Ch. 6 - A Pareto random variable X with parameters a0,0...Ch. 6 - Prob. 6.24STPE Ch. 6 - Prob. 6.25STPE Ch. 6 - Let X1,...,Xn, be independent nonnegative integer...

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Suppose that (X,Y) has a bivariate normal distribution with parameters μ x , μ y , σ x , σ , ρ . a. Show that ( X − μ x σ x , Y − μ y σ y ) has a bivariate normal distribution with parameters 0,1,0,1, ρ . b. What is the joint distribution of ( a X + b , c Y + d ) .

Solution Summary: The author explains the bivariate normal distribution of the given function. The joint density function mathrmfleft is a random variable.

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Chapter 6 Solutions