Problem 6.1P: Two fair dice are rolled. Find the joint probability mass function of X and Y when a. X is the... Problem 6.2P: Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8 red... Problem 6.3P: In Problem 8 t, suppose that the white balls are numbered, and let Yi equal 1 if the ith white ball... Problem 6.4P: Repeat Problem 6.2 when the ball selected is replaced in the urn before the next selection. Problem 6.5P: Repeat Problem 6.3a when the ball selected is replaced in the urn before the next selection. Problem 6.6P: The severity of a certain cancer is designated by one of the grades 1,2,3,4 with 1 being the least... Problem 6.7P: Consider a sequence of independent Bernoulli trials, each of which is a success with probability p.... Problem 6.8P Problem 6.9P: The joint probability density function of X and Y is given by f(x,y)=67(x2+xy2)0x1,0y2 a. Verify... Problem 6.10P Problem 6.11P: In Example Id, verify that f(x,y)=2exe2y,0x,0y, is indeed a joint density, function. That is, check... Problem 6.12P: The number of people who enter a drugstore in a given hour is a Poisson random variable with... Problem 6.13P: A man and a woman agree to meet at a certain location about 12:30 P.M. If the man arrives at a time... Problem 6.14P: An ambulance travels back and forth at a constant speed along a road of length L. At a certain... Problem 6.15P: The random vector (X,Y) is said to be uniformly distributed over a region R in the plane if, for... Problem 6.16P: Suppose that n points are independently chosen at random on the circumference of a circle, and we... Problem 6.17P Problem 6.18P: Let X1 and X2 be independent binomial random variables with Xi having parameters (ni,pi),i=1,2. Find... Problem 6.19P: Show that f(x,y)=1x, 0yx1 is a joint density function. Assuming that f is the joint density function... Problem 6.20P Problem 6.21P: Let f(x,y)=24xy0x1,0y1,0x+y1 and let it equal 0 otherwise. a. Show that f(xy) is a joint probability... Problem 6.22P: The joint density function of X and Y is f(x,y)={x+y0x1,0y10otherwise a. Are X and Y independent? b.... Problem 6.23P Problem 6.24P: Consider independent trials, each of which results in outcome i,i=0,1,...,k with probability... Problem 6.25P: Suppose that 106 people arrive at a service station at times that are independent random variables,... Problem 6.26P Problem 6.27P Problem 6.28P: The time that it takes to service a car is an exponential random variable with rate 1. a. If A. J.... Problem 6.29P: The gross daily sales at a certain restaurant are a normal random variable with mean $2200 and... Problem 6.30P: Jills bowling scores are approximately normally distributed with mean 170 and standard deviation 20,... Problem 6.31P: According to the U.S. National Center for Health Statistics, 25.2 percent of males and 23.6 percent... Problem 6.32P: Monthly sales are independent normal random variables with mean 100 and standard deviation a. Find... Problem 6.33P: Let X1 and X2 be independent normal random variables, each having mean 10 and variance 2. Which... Problem 6.34P Problem 6.35P: Teams 1, 2, 3, 4 are all scheduled to play each of the other teams 10 times. Whenever team i plays... Problem 6.36P: Let X1,...,X10 be independent with the same continuous distribution function F and let m be the... Problem 6.37P: The expected number of typographical errors on a page of a certain magazine is .2. What is the... Problem 6.38P: The monthly worldwide average number of airplane crashes of commercial airlines is 2.2. What is the... Problem 6.39P: In Problem 6.4, calculate the conditional probability mass function of X1 given that a. X2=1; b.... Problem 6.40P: In Problem 6.3 calculate the conditional probability mass function of Y1 given that a. Y2=1; b.... Problem 6.41P Problem 6.42P Problem 6.43P Problem 6.44P: The joint probability mass function of X and Y is given byP(1,1)=18p(1,2)=14P(2,1)=18p(2,2)=12 a.... Problem 6.45P Problem 6.46P Problem 6.47P: An insurance company supposes that each person has an accident parameter and that the yearly number... Problem 6.48P: If X1,X2,X3 are independent random variables that are uniformly distributed over (0, 1), compute the... Problem 6.49P Problem 6.50P: If 3 trucks break down at points randomly distributed on a road of length L find the probability... Problem 6.51P: Consider a sample of size 5 from a uniform distribution over (0, 1). Compute the probability that... Problem 6.52P Problem 6.53P: Let X(1),X(2),...,X(n) be the order statistics of a set of n independent uniform (0, 1) random... Problem 6.54P: Let Z1 and Z2 be independent standard normal random variables. Show that X, Y has a bivariate normal... Problem 6.55P: Derive the distribution of the range of a sample of size 2 from a distribution having density... Problem 6.56P: Let X and Y denote the coordinates of a point uniformly chosen in the circle of radius I centered at... Problem 6.57P Problem 6.58P Problem 6.59P Problem 6.60P Problem 6.61P: Repeat Problem 6.60 when X and Y are independent exponential random variables, each with parameter... Problem 6.62P Problem 6.63P Problem 6.64P: In Example 8b, let Yk+1=n+1i=1kYi. Show that Y1,....,Yk,Yk+1 are exchangeable. Note that Yk+1 is the... Problem 6.65P: Consider an urn containing n balls numbered 1.. .. . n, and suppose that k of them are randomly... Problem 6.1TE: Suppose X,Y have a joint distribution function F(x,y). Show how to obtain the distribution functions... Problem 6.2TE Problem 6.3TE Problem 6.4TE: Solve Buffons needle problem when LD. Problem 6.5TE: If X and Y are independent continuous positive random variables, express the density function of (a)... Problem 6.6TE Problem 6.7TE Problem 6.8TE: Let X and Y be independent continuous random variables with respective hazard rate functions X(t)... Problem 6.9TE: Let X1,...,Xn be independent exponential random variables having a common parameter . Determine the... Problem 6.10TE: The lifetimes of batteries are independent exponential random variables, each having parameter . A... Problem 6.11TE Problem 6.12TE: Show that the jointly continuous (discrete) random variables X1,...,Xn are independent if and only... Problem 6.13TE: In Example 5e t, we computed the conditional density of a success probability for a sequence of... Problem 6.14TE: Suppose that X and Y are independent geometric random variables with the same parameter a. Without... Problem 6.15TE: Consider a sequence of independent trials, with each trial being a success with probability p. Given... Problem 6.16TE: If X and Y are independent binomial random variables with identical parameters n and p. show... Problem 6.17TE: Suppose that Xi,i=1,2,3 are independent Poisson random variables with respective means i,i=1,2,3.... Problem 6.18TE Problem 6.19TE: Let X1,X2,X3 be independent and identically distributed continuous random variables. Compute a.... Problem 6.20TE Problem 6.21TE: Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with... Problem 6.22TE: Let W be a gamma random variable with parameters (t,) and suppose that conditional on W=,X1,X2,...Xn... Problem 6.23TE: A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to... Problem 6.24TE: If X is exponential with rate , find P{[X]=n,X[X]x} where [x] is defined as the largest integer less... Problem 6.25TE: Suppose thatF(x) is a cumulative distribution function. Show that (a) Fn(x) and (b)1[1F(x)]n are... Problem 6.26TE: Show that if n people are distributed at random along a road L miles long, then the probability that... Problem 6.27TE: Suppose that X1,...,Xn are independent exponential random variables with rate A. Find a.... Problem 6.28TE: Establish Equation (6.2) by differentiating Equation (6.4). Problem 6.29TE: Show that the median of a sample of size 2n+1 from a uniform distribution on (0, 1) has a beta... Problem 6.30TE Problem 6.31TE: Compute the density of the range of a sample of size n, from a continuous distribution having... Problem 6.32TE: Let X(1)X(2)...X(n) be the ordered values of n independent uniform (0, 1) random variables. Prove... Problem 6.33TE: Let X1,...,Xn be a set of independent and identically distributed continuous random variables having... Problem 6.34TE: Let X1,....Xn, be independent and identically distributed random variables having distribution... Problem 6.35TE Problem 6.36TE Problem 6.37TE: Suppose that (X,Y) has a bivariate normal distribution with parameters x,y,x,,. a. Show that... Problem 6.38TE: Suppose that X has a beta distribution with parameters (a, b) and that the conditional distribution... Problem 6.39TE: 6.39. Consider an experiment with n possible outcomes, having respective probabilities Problem 6.40TE Problem 6.41TE Problem 6.1STPE: Each throw of an unfair die lands on each of the odd numbers 1, 3, 5 with probability C and on each... Problem 6.2STPE: The joint probability mass function of the random variables X,Y,Z... Problem 6.3STPE Problem 6.4STPE: Let r=r1+...+rk, where all ri are positive integers. Argue that if X1,...,Xr has a multinomial... Problem 6.5STPE: Suppose that X, Y, and Z are independent random variables that are each equally likely to be either... Problem 6.6STPE: Let X and Y be continuous random variables with joint density functionf(x,y)={x5+cy0x1,1y50otherwise... Problem 6.7STPE: The joint density function of X and Y isf(x,y)={xy0x1,0y20otherwise a. Are X and Y independent? b.... Problem 6.8STPE: Consider two components and three types of shocks. A type I shock causes component I to fail, a type... Problem 6.9STPE: Consider a directory of classified advertisements that consists of m pages, where m is very large.... Problem 6.10STPE: The random parts of the algorithm in Self-Test Problem 6.9 &1 can be written in terms of the... Problem 6.11STPE Problem 6.12STPE: The accompanying dartboard is a square whose sides are of length 6: The three circles are all... Problem 6.13STPE: A model proposed for NBA basketball supposes that when two teams with roughly the same record play... Problem 6.14STPE: Let N be a geometric random variable with parameter p. Suppose that the conditional distribution of... Problem 6.15STPE Problem 6.16STPE: You and three other people are to place bids for an object, with the high bid winning. If you win,... Problem 6.17STPE: Find the probability that X1,X2,...,Xn is a permutation of 1, 2, …., n, when X1,X2,...,Xn are... Problem 6.18STPE: 6.18. Let 4VH and Y, be independent random vectors, with each vector being a random ordering of k... Problem 6.19STPE: Let Z1,Z2.....Zn be independent standard normal random variables, and let Sj=i=1jZi a. What is the... Problem 6.20STPE: Let X1,X2,... be a sequence of independent and identically distributed continuous random variables.... Problem 6.21STPE: Prove the identity P{Xs,Yt}=P{Xs}+P{Yt}+P{Xs,Yt}1 Hint: Derive an expression for P{Xs,Yt} by taking... Problem 6.22STPE: In Example 1c, find P(Xr=i,Ys=j) when ji. Problem 6.23STPE: A Pareto random variable X with parameters a0,0 has distribution functionF(x)=ax,xa, xa. For x0a,... Problem 6.24STPE Problem 6.25STPE Problem 6.26STPE: Let X1,...,Xn, be independent nonnegative integer valued random variables, and let... format_list_bulleted
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