Problem 7.1P: A player throws a fair die and simultaneously flips a fair coin, If the coin lands heads, then she... Problem 7.2P: The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is randomly chosen and the... Problem 7.3P: Gambles are independent, and each one results in the player being equally likely to win or lose 1... Problem 7.4P Problem 7.5P: The county hospital is located at the center of a square whose sides are 3 miles wide. If an... Problem 7.6P: A fair die is rolled 10 times. Calculate the expected sum of the 10 rolls. Problem 7.7P: Suppose that A and B each randomly and independently choose 3 of 10 objects. Find the expected... Problem 7.8P: N people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or... Problem 7.9P: A total of n. balls, numbered 1 through n, are put into n urns, also numbered 1 through n in such a... Problem 7.10P: Consider 3 trials, each having the same probability of success. Let X denote the total number of... Problem 7.11P: Consider n independent flips of a coin having probability p of landing on heads. Say that a... Problem 7.12P: A group of n men and n women is lined up at random. a. Find the expected number of men who have a... Problem 7.13P: A set of 1000 cards numbered 1 through 1000 is randomly distributed among 1000 people with each... Problem 7.14P: An urn has m black balls. At each stage, a black ball is removed and a new ball that is black with... Problem 7.15P: In Example 2h, say that i and j, ij form a matched pair if i chooses the hat belonging to j and j... Problem 7.16P: Let Z be a standard normal random variable, and, for a fixed x. set X={XifXx0otherwise Show that... Problem 7.17P: A deck of n cards numbered 1 through n is thoroughly shuffled so that all possible n! orderings can... Problem 7.18P: Cards from an ordinary deck of 52 playing cards are turned face up one at a time. If the 1st card is... Problem 7.19P Problem 7.20P Problem 7.21P: For a group of 100 people, compute a. the expected number of days of the year that are birthdays of... Problem 7.22P: How many times would you expect to roll a fair die before all 6 sides appeared at least once? Problem 7.23P: Urn I contains 5 white and 6 black balls, while urn 2 contains 8 white and 10 black balls. Two balls... Problem 7.24P: A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses... Problem 7.25P: Let X1,X2... be a sequence of independent and identically distributed continuous random variables.... Problem 7.26P: If X1,X2,....Xn are independent and identically distributed random variables having uniform... Problem 7.27P: If 101 items are distributed among 10 boxes, then at least one of the boxes must contain more than... Problem 7.28P Problem 7.29P: There are 4 different types of coupons, the first 2 of which comprise one group and the second 2... Problem 7.30P: If X and Y are independent and identically distributed with mean and variance2 find E[(XY)2] Problem 7.31P Problem 7.32P Problem 7.33P: If E[X]=1 and Var(X)=5, find a. E[(2+X)2]: b. Var(4+3X). Problem 7.34P: If 10 married couples are randomly seated at a round table, compute (a) the expected number and (b)... Problem 7.35P: Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards... Problem 7.36P: Let X be the number of ls and F the number of 2s that occur in n rolls of a fair die. Compute... Problem 7.37P: A die is rolled twice. Let X equal the sum of the outcomes, and let Y equal the first outcome minus... Problem 7.38P: Suppose X and Y have the following joint probability mass... Problem 7.39P: Suppose that 2 balls are randomly removed from an urn containing n red and m blue balls. Let Xi=1 if... Problem 7.40P Problem 7.41P: Let X1,... be independent with common mean and common variance 2 and setYn=Xn+Xn+1+Xn+2. For j0.... Problem 7.42P Problem 7.43P: A pond contains 100 fish, of which 30 are carp. If 20 fish are caught, what are the mean and... Problem 7.44P: A group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each.... Problem 7.45P: Let X1,X2,...,Xn be independent random variables having an unknown continuous distribution function... Problem 7.46P: Between two distinct methods for manufacturing certain goods, the quality of goods produced by... Problem 7.47P Problem 7.48P: Consider the following dice game. as played at a certain gambling casino: Players 1 and 2 roll a... Problem 7.49P Problem 7.50P: A fair die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary... Problem 7.51P: There are two misshapen coins in a box; their probabilities for landing on heads when they are... Problem 7.52P: The joint density of X and Y is given by f(x,y)=exyeyy,0x,0y Compute E[X2Y=y]. Problem 7.53P: The joint density of X and Y is given by f(x,y)=eyy,0xy,0y ComputeE[X3Y=y]. Problem 7.54P: A population is made up of r disjoint subgroups. Let pi denote the proportion of the population that... Problem 7.55P: A prisoner is trapped in a cell containing 3 doors. The first door leads to a tunnel that returns... Problem 7.56P: Consider the following dice game: A pair of dice is rolled. If the sum is 7, then the game ends and... Problem 7.57P: Ten hunters are waiting for ducks to fly by. When a flock of ducks flies overhead, the hunters fire... Problem 7.58P: The number of people who enter an elevator on the ground floor is a Poisson random variable with... Problem 7.59P: Suppose that the expected number of accidents per week at an industrial plant is 5. Suppose also... Problem 7.60P: A coin having probability p of coming up heads is continually flipped until both heads and tails... Problem 7.61P: A coin that comes up heads with probability p is continually flipped. Let N be the number of flips... Problem 7.62P: There are n+1 participants in a game. Each person independently is a winner with probability p. The... Problem 7.63P: Each of m+2 players pays 1 unit to a kitty in order to play the following game: A fair coin is to be... Problem 7.64P: The number of goals that J scores in soccer games that her team wins is Poisson distributed with... Problem 7.65P Problem 7.66P Problem 7.67P Problem 7.68P Problem 7.69P: Type i light bulbs function for a random amount of time having mean i, and standard deviation... Problem 7.70P: The number of winter storms in a good year is a Poisson random variable with mean 3, whereas the... Problem 7.71P: In Example 5c, compute the variance of the length of time until the miner reaches safety. Problem 7.72P Problem 7.73P: The number of accidents that a person has in a given year is a Poisson random variable with mean .... Problem 7.74P: Repeat Problem 7.73 when the proportion of the population having a value of less than x is equal to... Problem 7.75P: Consider an urn containing a large number of coins, and suppose that each of the coins has some... Problem 7.76P: In problem ,suppose that the coin is tossed n times. Let X denote the number of heads that occur.... Problem 7.77P: Suppose that in Problem 7.75, we continue to flip the coin until a head appears. Let N denote the... Problem 7.78P: In Example 6b, let S denote the signal sent and R the signal received. a. Compute E[R]. b. Compute... Problem 7.79P: In Example 6c y)2]. Problem 7.80P: The moment generating function of X is given by MX(t)=exp{2et2} and that of Y by MY(t)=(34et+14)10.... Problem 7.81P: Let X be the value of the first die and Y the sum of the values when two dice are rolled. Compute... Problem 7.82P: The joint density of X and Y is given by f(x,y)=12eye(xy)220y,x a. Compute the joint moment... Problem 7.83P Problem 7.84P: Successive weekly sales, in units of $1,000, have a bivariate normal distribution with common mean... Problem 7.1TE: Show that E[(Xa)2] is minimized at a=E[X]. Problem 7.2TE: Suppose that X is a continuous random variable with density function f. Show that E[|Xa|] is... Problem 7.3TE Problem 7.4TE: Let X be a random variable having finite expectation and variance 2 and let g(•) be a twice... Problem 7.5TE Problem 7.6TE Problem 7.7TE Problem 7.8TE: We say that X is stochastically larger than Y, written XstY, if, for all t, P{Xt}P{Yt} Show that if... Problem 7.9TE Problem 7.10TE: A coin having probability p of landing on heads is flipped n times. Compute the expected number of... Problem 7.11TE: Let X1,X2,....Xn be independent and identically distributed positive random variables. For kn, find... Problem 7.12TE Problem 7.13TE: Let X1,X2,... be a sequence of independent random variables having the probability mass function... Problem 7.14TE Problem 7.15TE Problem 7.16TE Problem 7.17TE Problem 7.18TE Problem 7.19TE: In Example 41 t, we showed that the covariance of the multinomial random variables Ni and Nj is... Problem 7.20TE: Show that X and Y are identically distributed and not necessarily independent, then cov(X+Y,XY)=0 Problem 7.21TE Problem 7.22TE Problem 7.23TE Problem 7.24TE: Show that Z is a standard normal random variable and if Y is defined by Y=a+bZ+eZ2, then... Problem 7.25TE: Prove the Cauchy-Schwarz inequality, namely, (E[XY])2E[X2]E[Y2] Hint: Unless Y=tX for some constant,... Problem 7.26TE: Show that if X and Y are independent, then E[XY=y]=E[X] for all y a. in the discrete case; b. in the... Problem 7.27TE: Prove that E[g(X)YX]=g(X)E[YX]. Problem 7.28TE: Prove that if E[YX=x]=E[Y] for all x, then X and Y are uncorrelated; give a counterexample to show... Problem 7.29TE Problem 7.30TE: Let X1,...,Xn be independent and identically distributed random variables, FindE[X1X1+...+Xn=x] Problem 7.31TE: Consider Example 4f, which is concerned with the multinomial distribution. Use conditional... Problem 7.32TE: An urn initially contains b black and w white balls. At each stage, we add r black balls and then... Problem 7.33TE: For an event A, let IA equal 1 if A occurs and let it equal 0 if A does not occur. For a random... Problem 7.34TE: A coin that lands on heads with probability p is continually flipped. Compute the expected number of... Problem 7.35TE: For another approach to Theoretical Exercise 7.34, let Tr denote the number of flips required to... Problem 7.36TE: The probability generating function of the discrete nonnegative integer valued random variable X... Problem 7.37TE: One ball at a time is randomly selected from an urn containing a white and b black balls until all... Problem 7.38TE Problem 7.39TE Problem 7.40TE: The best quadratic predictor of Y with respect to X is a+bX+cX2, where a, b, and c are chosen to... Problem 7.41TE: Use the conditional variance formula to determine the variance of a geometric random variable X... Problem 7.42TE: Let X be a normal random variable with parameters =0 and 2=1, and let I, independent of X, be such... Problem 7.43TE: It follows from Proposition 6.1 and the fact that the best linear predictor of Y with respect to... Problem 7.44TE: Show that for random variables X and Z, E[(XY)2]=E(X2)E(Y2) where Y=[XZ] Problem 7.45TE Problem 7.46TE: Verify the formula for the moment generating function of a uniform random variable that is given in... Problem 7.47TE: For a standard normal random variable Z, let n=E[Zn]. Show thatn={0whennisodd(2j)!2jj!whenn=2j Hint:... Problem 7.48TE Problem 7.49TE Problem 7.50TE: The positive random variable X is said to be a lognormal random variable with parameters and 2 if... Problem 7.51TE: Let X have moment generating function M(t), and define (t)=logM(t). Show that (t)t=0=var(X) Problem 7.52TE: Use Table 7.2 to determine the distribution of i=1nXi when X1,...,Xn are independent and identically... Problem 7.53TE: Show how to compute cov(X,Y) from the joint moment generating function of X and Y. Problem 7.54TE: Suppose that X1,...,Xn have a multivariate normal distribution. Show that X1,...,Xn are independent... Problem 7.55TE: If Z is a standard normal random variable, what is cov(Z,Z2)? Problem 7.56TE: Suppose that Y is a normal random variable with mean and variance 2, and suppose also that the... Problem 7.1STPE: Consider a list of m names, where the same name may appear more than once on the list. Let... Problem 7.2STPE Problem 7.3STPE Problem 7.4STPE Problem 7.5STPE Problem 7.6STPE Problem 7.7STPE Problem 7.8STPE Problem 7.9STPE Problem 7.10STPE Problem 7.11STPE Problem 7.12STPE Problem 7.13STPE Problem 7.14STPE Problem 7.15STPE Problem 7.16STPE Problem 7.17STPE Problem 7.18STPE Problem 7.19STPE: There are n items in a box labeled H and m in a box labeled T. A coin that comes up heads with... Problem 7.20STPE: Let X be a nonnegative random variable having distribution function F. Show that if F(x)=1F(x) ,then... Problem 7.21STPE: Let a1,...,an, not all equal to 0, be such that i=1na=0. Show that there is a permutation i1,...,in... Problem 7.22STPE Problem 7.23STPE Problem 7.24STPE Problem 7.25STPE Problem 7.26STPE Problem 7.27STPE Problem 7.28STPE Problem 7.29STPE Problem 7.30STPE Problem 7.31STPE Problem 7.32STPE Problem 7.33STPE format_list_bulleted
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,