A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 7, 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
NOTE: The preceding has possible applications to understanding the AIDS disease. Part of the body’s immune system consists of a certain class of cells known as T-cells. There are 2 types of T-cells, called CD4 and CD8. Now, while the total number of T-cells in AIDS sufferers is (at least in the early stages of the disease) the same as that in healthy individuals, it has recently been discovered that the mix of CD4 and CD8 T-cells is different. Roughly 60 percent of the T-cells of a healthy person are of the CD4 type, whereas the percentage of the T-cells that are of CD4 type appears to decrease continually in AIDS sufferers. A recent model proposes that the HIV virus (the virus that causes AIDS) attacks CD4 cells and that the body’s mechanism for replacing killed T-cells does not differentiate between whether the killed T-cell was CD4 or CD8. Instead, it just produces a new T-cell that is CD4 with probability .6 and CD8 with probability .4. However, although this would seem to be a very efficient way of replacing killed T-cells when each one killed is equally likely to be any of the body’s T-cells (and thus has probability .6 of being CD4), it has dangerous consequences when facing a virus that targets only the CD4 T-cells.
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Chapter 7 Solutions
A First Course in Probability (10th Edition)
Ch. 7 - A player throws a fair die and simultaneously...Ch. 7 - The game of Clue involves 6 suspects, 6 weapons,...Ch. 7 - Gambles are independent, and each one results in...Ch. 7 - Prob. 7.4PCh. 7 - The county hospital is located at the center of a...Ch. 7 - A fair die is rolled 10 times. Calculate the...Ch. 7 - Suppose that A and B each randomly and...Ch. 7 - N people arrive separately to a professional...Ch. 7 - A total of n. balls, numbered 1 through n, are put...Ch. 7 - Consider 3 trials, each having the same...
Ch. 7 - Consider n independent flips of a coin having...Ch. 7 - A group of n men and n women is lined up at...Ch. 7 - A set of 1000 cards numbered 1 through 1000 is...Ch. 7 - An urn has m black balls. At each stage, a black...Ch. 7 - In Example 2h, say that i and j, ij form a matched...Ch. 7 - Let Z be a standard normal random variable, and,...Ch. 7 - A deck of n cards numbered 1 through n is...Ch. 7 - Cards from an ordinary deck of 52 playing cards...Ch. 7 - Prob. 7.19PCh. 7 - Prob. 7.20PCh. 7 - For a group of 100 people, compute a. the expected...Ch. 7 - How many times would you expect to roll a fair die...Ch. 7 - Urn I contains 5 white and 6 black balls, while...Ch. 7 - A bottle initially contains m large pills and n...Ch. 7 - Let X1,X2... be a sequence of independent and...Ch. 7 - If X1,X2,....Xn are independent and identically...Ch. 7 - If 101 items are distributed among 10 boxes, then...Ch. 7 - Prob. 7.28PCh. 7 - There are 4 different types of coupons, the first...Ch. 7 - If X and Y are independent and identically...Ch. 7 - Prob. 7.31PCh. 7 - Prob. 7.32PCh. 7 - If E[X]=1 and Var(X)=5, find a. E[(2+X)2]: b....Ch. 7 - If 10 married couples are randomly seated at a...Ch. 7 - Cards from an ordinary deck are turned face up one...Ch. 7 - Let X be the number of ls and F the number of 2s...Ch. 7 - A die is rolled twice. Let X equal the sum of the...Ch. 7 - Suppose X and Y have the following joint...Ch. 7 - Suppose that 2 balls are randomly removed from an...Ch. 7 - Prob. 7.40PCh. 7 - Let X1,... be independent with common mean and...Ch. 7 - Prob. 7.42PCh. 7 - A pond contains 100 fish, of which 30 are carp. If...Ch. 7 - A group of 20 people consisting of 10 men and 10...Ch. 7 - Let X1,X2,...,Xn be independent random variables...Ch. 7 - Between two distinct methods for manufacturing...Ch. 7 - Prob. 7.47PCh. 7 - Consider the following dice game. as played at a...Ch. 7 - Prob. 7.49PCh. 7 - A fair die is successively rolled. Let X and Y...Ch. 7 - There are two misshapen coins in a box; their...Ch. 7 - The joint density of X and Y is given by...Ch. 7 - The joint density of X and Y is given by...Ch. 7 - A population is made up of r disjoint subgroups....Ch. 7 - A prisoner is trapped in a cell containing 3...Ch. 7 - Consider the following dice game: A pair of dice...Ch. 7 - Ten hunters are waiting for ducks to fly by. When...Ch. 7 - The number of people who enter an elevator on the...Ch. 7 - Suppose that the expected number of accidents per...Ch. 7 - A coin having probability p of coming up heads is...Ch. 7 - A coin that comes up heads with probability p is...Ch. 7 - There are n+1 participants in a game. Each person...Ch. 7 - Each of m+2 players pays 1 unit to a kitty in...Ch. 7 - The number of goals that J scores in soccer games...Ch. 7 - Prob. 7.65PCh. 7 - Prob. 7.66PCh. 7 - Prob. 7.67PCh. 7 - Prob. 7.68PCh. 7 - Type i light bulbs function for a random amount of...Ch. 7 - The number of winter storms in a good year is a...Ch. 7 - In Example 5c, compute the variance of the length...Ch. 7 - Prob. 7.72PCh. 7 - The number of accidents that a person has in a...Ch. 7 - Repeat Problem 7.73 when the proportion of the...Ch. 7 - Consider an urn containing a large number of...Ch. 7 - In problem ,suppose that the coin is tossed n...Ch. 7 - Suppose that in Problem 7.75, we continue to flip...Ch. 7 - In Example 6b, let S denote the signal sent and R...Ch. 7 - In Example 6c y)2].Ch. 7 - The moment generating function of X is given by...Ch. 7 - Let X be the value of the first die and Y the sum...Ch. 7 - The joint density of X and Y is given by...Ch. 7 - Prob. 7.83PCh. 7 - Successive weekly sales, in units of $1,000, have...Ch. 7 - Show that E[(Xa)2] is minimized at a=E[X].Ch. 7 - Suppose that X is a continuous random variable...Ch. 7 - Prob. 7.3TECh. 7 - Let X be a random variable having finite...Ch. 7 - Prob. 7.5TECh. 7 - Prob. 7.6TECh. 7 - Prob. 7.7TECh. 7 - We say that X is stochastically larger than Y,...Ch. 7 - Prob. 7.9TECh. 7 - A coin having probability p of landing on heads is...Ch. 7 - Let X1,X2,....Xn be independent and identically...Ch. 7 - Prob. 7.12TECh. 7 - Let X1,X2,... be a sequence of independent random...Ch. 7 - Prob. 7.14TECh. 7 - Prob. 7.15TECh. 7 - Prob. 7.16TECh. 7 - Prob. 7.17TECh. 7 - Prob. 7.18TECh. 7 - In Example 41 t, we showed that the covariance of...Ch. 7 - Show that X and Y are identically distributed and...Ch. 7 - Prob. 7.21TECh. 7 - Prob. 7.22TECh. 7 - Prob. 7.23TECh. 7 - Show that Z is a standard normal random variable...Ch. 7 - Prove the Cauchy-Schwarz inequality, namely,...Ch. 7 - Show that if X and Y are independent, then...Ch. 7 - Prove that E[g(X)YX]=g(X)E[YX].Ch. 7 - Prove that if E[YX=x]=E[Y] for all x, then X and Y...Ch. 7 - Prob. 7.29TECh. 7 - Let X1,...,Xn be independent and identically...Ch. 7 - Consider Example 4f, which is concerned with the...Ch. 7 - An urn initially contains b black and w white...Ch. 7 - For an event A, let IA equal 1 if A occurs and let...Ch. 7 - A coin that lands on heads with probability p is...Ch. 7 - For another approach to Theoretical Exercise 7.34,...Ch. 7 - The probability generating function of the...Ch. 7 - One ball at a time is randomly selected from an...Ch. 7 - Prob. 7.38TECh. 7 - Prob. 7.39TECh. 7 - The best quadratic predictor of Y with respect to...Ch. 7 - Use the conditional variance formula to determine...Ch. 7 - Let X be a normal random variable with parameters...Ch. 7 - It follows from Proposition 6.1 and the fact that...Ch. 7 - Show that for random variables X and Z,...Ch. 7 - Prob. 7.45TECh. 7 - Verify the formula for the moment generating...Ch. 7 - For a standard normal random variable Z, let...Ch. 7 - Prob. 7.48TECh. 7 - Prob. 7.49TECh. 7 - The positive random variable X is said to be a...Ch. 7 - Let X have moment generating function M(t), and...Ch. 7 - Use Table 7.2 to determine the distribution of...Ch. 7 - Show how to compute cov(X,Y) from the joint moment...Ch. 7 - Suppose that X1,...,Xn have a multivariate normal...Ch. 7 - If Z is a standard normal random variable, what is...Ch. 7 - Suppose that Y is a normal random variable with...Ch. 7 - Consider a list of m names, where the same name...Ch. 7 - Prob. 7.2STPECh. 7 - Prob. 7.3STPECh. 7 - Prob. 7.4STPECh. 7 - Prob. 7.5STPECh. 7 - Prob. 7.6STPECh. 7 - Prob. 7.7STPECh. 7 - Prob. 7.8STPECh. 7 - Prob. 7.9STPECh. 7 - Prob. 7.10STPECh. 7 - Prob. 7.11STPECh. 7 - Prob. 7.12STPECh. 7 - Prob. 7.13STPECh. 7 - Prob. 7.14STPECh. 7 - Prob. 7.15STPECh. 7 - Prob. 7.16STPECh. 7 - Prob. 7.17STPECh. 7 - Prob. 7.18STPECh. 7 - There are n items in a box labeled H and m in a...Ch. 7 - Let X be a nonnegative random variable having...Ch. 7 - Let a1,...,an, not all equal to 0, be such that...Ch. 7 - Prob. 7.22STPECh. 7 - Prob. 7.23STPECh. 7 - Prob. 7.24STPECh. 7 - Prob. 7.25STPECh. 7 - Prob. 7.26STPECh. 7 - Prob. 7.27STPECh. 7 - Prob. 7.28STPECh. 7 - Prob. 7.29STPECh. 7 - Prob. 7.30STPECh. 7 - Prob. 7.31STPECh. 7 - Prob. 7.32STPECh. 7 - Prob. 7.33STPE
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- Total marks 14 4. Let X and Y be random variables on a probability space (N, F, P) that take values in [0, ∞). Assume that the joint density function of X and Y on [0, ∞) × [0, ∞) is given by f(x, y) = 2e-2x-y Find the probability P(0 ≤ X ≤ 1,0 ≤ y ≤ 2). (ii) spectively. [6 Marks] Find the the probability density function of X and Y, re- [5 Marks] 111) Are the X and Y independent? Justify your answer! [3 Marks]arrow_forwardTotal marks 17 4. Let (,,P) be a probability space and let X : → R be a ran- dom variable that has Gamma(2, 1) distribution, i.e., the distribution of the random variable X is the probability measure on ((0, ∞), B((0, ∞))) given by (i) dPx(x) = xex dx. Find the characteristic function of the random variable X. [8 Marks] (ii) Using the result of (i), calculate the first three moments of the random variable X, i.e., E(X") for n = 1, 2, 3. Using Markov's inequality involving E(X³), (iii) probability P(X > 10). [6 Marks] estimate the [3 Marks]arrow_forward1. There are 8 balls in an urn, of which 6 balls are red, 1 ball is blue and 1 ball is white. You draw a ball from the urn at random, note its colour, do not return the ball to the urn, and then draw a second ball, note its colour, do not return the ball to the urn, and finally draw a third ball, note its colour. (i) (Q, F, P). Describe the corresponding discrete probability space [7 Marks] (ii) Consider the following event, A: At least one of the first two balls is red.arrow_forward
- 3. Consider the following discrete probability space. Let = {aaa, bbb, ccc, abc, acb, bac, bca, cab, cba}, i.e., consists of 3-letter 'words' aaa, bbb, ccc, and all six possible 3-letter 'words' that have a single letter a, a single letter b, and a single letter c. The probability measure P is given by 1 P(w) = for each weΩ. 9 Consider the following events: A: the first letter of a 'word' is a, B: the second letter of a 'word' is a, C: the third letter of a 'word' is a. answer! Decide whether the statements bellow are true or false. Justify your (i) The events A, B, C are pairwise independent. (ii) The events A, B, C are independent. Total marks 7 [7 Marks]arrow_forwardLet X and Y have the following joint probability density function: fxy(x,y) =1/(x²²), for >>1, y>1 0, otherwise Let U = 5XY and V = 3 x. In all question parts below, give your answers to three decimal places (where appropriate). (a) The non-zero part of the joint probability density function of U and V is given by fu,v(u,v) = A√³uc for some constants A, B, C. Find the value of A. Answer: 5 Question 5 Answer saved Flag question Find the value of B. Answer: -1 Question 6 Answer saved P Flag question (b) The support of (U,V), namely the values of u and vthat correspond to the non-zero part of fu,v(u,v) given in part (a), is given by:arrow_forwardTotal marks 13. 3. There are three urns. Urn I contains 3 blue balls and 5 white balls; urn II contains 2 blue balls and 6 white balls; urn III contains 4 blue balls and 4 white balls. Rolling a dice, if 1 appears, we draw a ball from urn I; if 4 or 5 or 6 appears, we draw a ball from urn II; if 2, or 3 appears, we draw a ball from urn III. (i) What is the probability to draw a blue ball? [7 Marks] (ii) Assume that a blue ball is drawn. What is the probability that it came from Urn I? [6 Marks] Turn over. MA-252: Page 3 of 4arrow_forward
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