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introduction to probability statistics
Introduction To Probability Volume 2 1st Edition Narayanaswamy Balakrishnan, Markos V. Koutras, Konstadinos G. Politis - Solutions
*45. An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn c additional balls of the same color are put in with it. Now suppose that we draw another ball. Show that the probability that the first ball drawn was black given that the
44. Urn 1 has five white and seven black balls. Urn 2 has three white and twelve black balls. We flip a fair coin. If the outcome is heads, then a ball from urn 1 is selected, while if the outcome is tails, then a ball from urn 2 is selected. Suppose that a white ball is selected.What is the
43. The blue-eyed gene for eye color is recessive, meaning that both the eye genes of an individual must be blue for that individual to be blue eyed. Jo (F) and Joe (M) are both brown-eyed individuals whose mothers had blue eyes. Their daughter Flo, who has brown eyes, is expecting a child
42. There are three coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the three coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?
41. In a certain species of rats, black dominates over brown. Suppose that a black rat with two black parents has a brown sibling.(a) What is the probability that this rat is a pure black rat(as opposed to being a hybrid with one black and one brown gene)?(b) Suppose that when the black rat is
*40.(a) A gambler has in his pocket a fair coin and a twoheaded coin. He selects one of the coins at random, and when he flips it, it shows heads. What is the probability that it is the fair coin?(b) Suppose that he flips the same coin a second time and again it shows heads. Now what is the
39. Stores , and C have 50, 75, and 100 employees, and, respectively, 50, 60, and 70 percent of these are women.Resignations are equally likely among all employees, regardless of sex. One employee resigns and this is a woman. What is the probability that she works in store C?
38. Urn 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in urn 2. A ball is then drawn from urn 2. It happens to be white. What is the probability that the transferred ball was white?
37. In Exercise 36, what is the probability that the first box was the one selected given that the marble is white?
36. Consider two boxes, one containing one black and one white marble, the other, two black and one white marble. A box is selected at random and a marble is drawn at random from the selected box. What is the probability that the marble is black?
35. A fair coin is continually flipped. What is the probability that the first four flips are(a) ?(b) ?(c) What is the probability that the pattern occurs before the pattern ?
34. There is a 40 percent chance that A can fix her busted computer. If A cannot, then there is a 20 percent chance that her friend B can fix it. Find the probability it will be fixed by either A or B.
33. The winner of a tennis match is the first player to win 2 sets. A golden set occurs when one of the players wins all 24 points of a set. Supposing that the results of successive points are independent and that each point is equally likely to be won by either player, find the probability that at
*32. Suppose all n men at a party throw their hats in the center of the room. Each man then randomly selects a hat. Show that the probability that none of the n men selects his own hat is Note that as this converges to . Is this surprising?
31. What is the conditional probability that the first die is six given that the sum of the dice is seven?
*30. Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability 0.7, whereas George, independently, hits the target with probability 0.4.(a) Given that exactly one shot hit the target, what is the probability that it was
29. Suppose that . What can you say about when(a) E and F are mutually exclusive?(b) ?(c) ?
28. If the occurrence of B makes A more likely, does the occurrence of A make B more likely?
*27. Suppose in Exercise 26 we had defined the events , by Now use Exercise 23 to find , the probability that each pile has an ace. Compare your answer with the one you obtained in Exercise 26.
26. A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards each. Define events , and as follows:
*25. Two cards are randomly selected from a deck of 52 playing cards.(a) What is the probability they constitute a pair (that is, that they are of the same denomination)?(b) What is the conditional probability they constitute a pair given that they are of different suits?
24. In an election, candidate A receives n votes and candidate B receives m votes, where . Assume that in the count of the votes all possible orderings of the votes are equally likely. Let denote the probability that from the first vote on A is always in the lead. Find
23. For events show that
22. A and B play until one has 2 more points than the other. Assuming that each point is independently won by A with probability p, what is the probability they will play a total of 2n points? What is the probability that A will win?
21. Suppose that 5 percent of men and 0.25 percent of women are colorblind. A randomly chosen person is colorblind. What is the probability of this person being male? Assume that there are an equal number of males and females.
20. Three dice are thrown. What is the probability the same number appears on exactly two of the three dice?
*19. Two dice are rolled. What is the probability that at least one is a six? If the two faces are different, what is the probability that at least one is a six?
18. Assume that each child who is born is equally likely to be a boy or a girl. If a family has two children, what is the probability that both are girls given that (a) the eldest is a girl, (b) at least one is a girl?
*17. Suppose each of three persons tosses a coin. If the outcome of one of the tosses differs from the other outcomes, then the game ends. If not, then the persons start over and retoss their coins.Assuming fair coins, what is the probability that the game will end with the first round of tosses?
16. Use Exercise 15 to show that.
15. Argue that .
14. The probability of winning on a single toss of the dice is p. A starts, and if he fails, he passes the dice to B, who then attempts to win on her toss. They continue tossing the dice back and forth until one of them wins. What are their respective probabilities of winning?
13. The dice game craps is played as follows. The player throws two dice, and if the sum is seven or eleven, then she wins. If the sum is two, three, or twelve, then she loses. If the sum is anything else, then she continues throwing until she either throws that number again (in which case she
56. For the logistics regression model, find the value x such that p(x) = .5
50. Explain why, for the same data, a prediction interval for a future response always contains the corresponding confidence interval for the mean response.
40. Redo Problem 5 under the assumption that the variance of the gain in reading speed is proportional to the number of weeks in the program.
36. The following data represent the bacterial count of five individuals at different times after being inoculated by a vaccine consisting of the bacteria(a) Fit a curve.(b) Estimate the bacteria count of a new patient after 8 days.
24. Plot the standardized residuals from the data of Problem 1. What does the plot indicate about the assumptions of the linear regression model?
23. (a) Estimate the variances in Problems 19 through 22.(b) Determine a 95 percent confidence interval for the variance in the data relating to lung cancer.(c) Break up the lung cancer data into two parts — the first corresponding to states whose average cigarette consumption is less than 2,300,
22. (a) Draw a scatter diagram of cigarettes smoked versus death rate from leukemia.(b) Estimate the regression coefficients.(c) Test the hypothesis that there is no regression of the death rate from leukemia on the number of cigarettes used. That is, test that β = 0.(d) Determine a 90 percent
21. (a) Draw a scatter diagram of cigarette use versus death rate from kidney cancer.(b) Estimate the regression line.(c) What is the p-value in the test that the slope of the regression line is 0?(d) Determine a 90 percent confidence interval for the mean death rate from kidney cancer in a state
20. (a) Draw a scatter diagram relating cigarette use and death rates from lung cancer.(b) Estimate the regression parameters α and β.(c) Test at the .05 level of significance the hypothesis that cigarette consumption does not affect the death rate from lung cancer.(d) What is the p-value of the
19. (a) Draw a scatter diagram of cigarette consumption versus death rate from bladder cancer.(b) Does the diagram indicate the possibility of a linear relationship?(c) Find the best linear fit.(d) If next year’s average cigarette consumption is 2,500, what is your prediction of the death rate
16. Verify Equation 9.4.3.
15. Experienced flight instructors have claimed that praise for an exceptionally fine landing is typically followed by a poorer landing on the next attempt, whereas criticism of a faulty landing is typically followed by an improved landing. Should we thus conclude that verbal praise tends to lower
9. In Problem 4,(a) Estimate the variance of an individual response.(b) Determine a 90 percent confidence interval for the variance.
A steel company is planning to produce cold reduced sheet steel consisting of .15 percent copper at an annealing temperature of 1,150 (degrees F), and is interested in estimating the average (Rockwell 30-T) hardness of a sheet. To determine this, they have collected the data shown in Table 9.6 on
For the data of Example 9.10a, we computed that SSR = 34.12. Since n = 8, k = 2, the estimate of σ2 is 34.12/5 = 6.824.
In Example 9.4c, suppose we want an interval that we can “be 95 percent certain” will contain the height of a given male whose father is 68 inches tall. A simple computation now yields the prediction interval Y (68) ∈ 67.568 ± 1.050 or, with 95 percent confidence, the person’s height will
Using the data of Example 9.4c, determine a 95 percent confidence interval for the average height of all males whose fathers are 68 inches tall.
Derive a 95 percent confidence interval estimate of β in Example 9.4a.
Jimmy plays a game for which if he wins, he receives 2c dollars and if he loses, he pays c dollars. If his probability of winning the game is 3∕5, and his expected earnings from this game are $2, then the value of c is(a) 5∕2 (b)3∕2 (c)6∕5 (d) 2 (e) 5
Assume that for a variable X, we have????X = 1∕2, E(X) = 0.The value of E(6X2 + 5) equals(a) 3∕2 (b) 13∕2 (c) 13 (d) 3 (e) 8
For the variable X, it is known that E(2X − 4) = 4, E(4X2 − 3) = 71.The variance of X equals(a) 7∕2 (b) 1 (c) 55 (d) 21∕2 (e)5∕2
When Nicholas tosses a coin, he has a probability p of getting heads and a probability q = 1 − p of getting tails. In a single coin toss, let X denote the number of heads. If Var(X) = 3∕16, then the value of p is(a) 1∕2 (b) 1∕4 (c) 3∕4(d) either 1∕4 or 3∕4 (e) either 1∕2 or 3∕4
Let f be a probability function of a random variable X defined by f (x) = cx, RX = {1, 2, 3}.Then, the expected value of X equals(a) 7∕6 (b)7∕3 (c)1∕6 (d)7∕2 (e) 14∕9
For the random variable X, we know that P(X = 1) = 0.3, P(X = 2) = 0.2, P(X = 5) = 0.5.Then, the expected value of X equals(a) 8∕3 (b) 2 (c) 1 (d) 3.2 (e) 2.7
If the distribution function F of a variable X has a jump at the pointa, then(a) X is a discrete random variable (b) X is a continuous random variable(c) P(X =a) = F(a) (d) P(X =a) =0 (e) P(X =a) > 0
Andrea throws a die three times and let X be the number of sixes that appear. The range of values for X is(a) RX = {1, 2, 3} (b) RX = {1, 2, 3, 4, 5, 6} (c) RX = {0, 1, 2, 3}(d) RX = {0, 1, 2, 3, 4, 5, 6} (e) none of the above
Then the probability P(X = 3) equals(a) 3∕10 (b) 1∕10 (c) 1∕4 (d)2∕5 (e)3∕5
X is a random variable that takes only the values 1, 2, 3, and 4 in a way such that the probability of the event {X = x} is proportional to x for x = 1, 2, 3,
Jimmy plays a game for which he wins $4 with probability 2∕3, or else he wins $8.Let X be his profit for this game. Then, an upper bound for the probability P(X ≥ 6)from Markov’s inequality is 8∕9.
Then X can only take one value, i.e. there exists a c such that P(X =c) = 1.
Let X be a random variable with E(X) = 3 and E(X2) =
If a random variable X takes two values x1, x2 with x1 ≠ x2, then Var(X) > [E(X)]2.
Then for the random variable Y = 3X2 − 5X, we have E(Y) = 1.
For the random variable X, it is known that E(X) = 1 and Var(X) =
When we toss a coin twice, the variance for the number of tails that appear is 1∕2.
Let X be the number of heads when we toss a coin twice. Then E(X2) = 3∕2.
When we toss a coin three times, the expected number of heads that appear is 3∕2.
For any random variable X, the standard deviation of X is greater than or equal to its expected value.
For any random variable X, we have Var(X) ≥ E(X).
Let a function f be such that f (x) = x − 1 5 .Then f can be the probability function of a random variable X with range{0, 1, 2, 3, 4}.
If X is a discrete random variable with distribution function F, then for any a and b, P(a ≤ X ≤b) = F(b) − F(a).
If a continuous random variable X has distribution function F, then for any a and b, P(a < X
X is a discrete random variable with distribution function F. Then for any real a, we have P(a < X) = F(a).
Let f be a probability function of a random variable X. Then we have that limx→∞f (x) = 1.
The probability function of a random variable is always a nondecreasing function.
We throw a die repeatedly, and let Y be the number of throws until 5 appears for the first time. Then the range of values for Y is {0, 1, 2, 3, 4, 6}.
When we toss a coin five times, and X denotes the number of heads that appear, then the range of values for X is {0, 1, 2, 3, 4, 5}.
With the following program, we study the distribution of the sum of outcomes when two dice are thrown in succession. More specifically, we find the probability function, the expected value and the variance of that distribution, and we draw the graph of the probability function. Note that the values
We choose at random one ball from each urn.We want to find the probability functions of the variables X and Y defined as follows:X: the larger of the two numbers in the balls drawn, Y: the smaller of the two numbers in the balls drawn.
The next program calculates the probability function for the number of times, X, that the ordered pair TH appears when tossing four coins successively (here, H has been coded as “1” and T has been coded as “0”):In[1]:= n1=1;n2=1;n3=1;n4=1;Print["List of all outcomes of the
Recall the simulation program of Section 1.8. By suitably modifying this program, find an approximation for the expected profit per game for a player who participates in each of the following games:(i) We throw two dice simultaneously. If exactly one die lands on a 6, we win $10, if both dice land
Let f be the probability function of a discrete random variable X with range{m,m + 1,m + 2,…, n}, and let g be a function defined on the same set. Write a program to calculate the expectation E[g(X)]. As an application,(i) solve Example 4.17;(ii) carry out the calculations necessary in Exercise
Let us consider a discrete random variable X with a probability function of the form f (x) = cx2, x ∈ RX = {1, 2,…, n}, for a suitable constantc. The following sequence of commands enables us to find the value ofc, and subsequently the mean and variance of X.In[1]:= f1[x_]:=x ̂ 2 a=Sum[f1[x],
For the discrete random variable X, suppose that E(X) = Var(X) = ????.What can you infer from Chebyshev’s inequality for the probability P(X ≥ 2????)?
For a nonnegative discrete random variable X, we know that E(X) = 25, E[X(X − 4)] = 900.Find an upper bound for the probability P(X ≥ 50) using(i) Markov’s inequality;(ii) Chebyshev’s inequality.
The number of mail items handled daily by a courier service is a random variable. It has been estimated that this variable has a mean of 3000 items and variance 40 000.Obtain a lower bound for the probability that, in a given day, the company handles more than 2400 but less than 3600 mail items.
Explain why this is true (no calculations are needed).
Consider now two random variables X and Y having the same range RX = RY = {a1, a2,…, an}, and assume that E(Xr) = E(Yr) for r = 1, 2,…, n − 1.Arguing as in the previous exercise, verify again that the distributions of the variables X and Y are identical.
Let X be a random variable and t be a real number such that the moment E[(X − t)2]exists.(i) Verify that E[(X − t)2] = E(X2) − 2tE(X) + t2.(ii) Show that E[(X − t)2] = Var(X) + (???? − t)2.(iii) Use the result in (ii) to establish that the minimum value of the function h(t) = E[(X −
Walter throws two dice. If the sum of the two outcomes is 10 or more, he receives$5. If the sum is 8 or 9, nothing happens, while if the sum is less than 8, he loses c dollars. If X denotes Walter’s profit from this game, find the variance of X given that the game is fair.
Let for r = 1, 2,…,????(r) = E[X(X − 1) · · · (X − r + 1)]be the factorial moments of the random variable X, and????r = [E(X − ????)r], ????′r = E(Xr), r = 1, 2,…, be the moments of X around its mean, ???? = E(X) (central moments), and around zero, respectively.Using properties of
Suppose that a discrete random variable X has probability function f (x) = P(X = x) = 1 5, x ∈ RX = {−2a,−a, 0,a, 2a}.(i) Show that, regardless of the value ofa, we have E(X) = 0.(ii) Calculate Var(X) as a function ofa. What happens with the variance as a increases? Give an intuitive
Verify the truth of (4.12) in Example 4.25 for any discrete random variable X and any positive integer r.
Calculate Var(2X + 4) if it is known that E(5X − 1) = 9 and E[(X + 2)2] = 17.
If, for a variable X it is known that Var(2018 − X) = 2, find Var(−5X + 1).
The probability function of a discrete random variable is given by f (x) = c(32 − x), x = 1, 2,…, 31, where c is a constant.(i) Show that c = 1∕496.(ii) Find the expectation E(X).(iii) Find the expectation of the random variable Y = 3(X − 11) and show that Var(Y) = E(Y2).
With reference to Exercise 5, Section 4.4 wherein Jimmy participates in a quiz show with multiple choice questions:(i) Calculate the variance for the number of correct answers that he gives.(ii) Calculate the variance and standard deviation of Jimmy’s earnings from the show.
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