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Questions and Answers of Statistics

Let Ω = {a, b, c, d, e, f}. Assume that m(a) = m(b) = 1/8 and m(c) = m(d) = m(e) = m(f) = 3/16. Let A, B, and C be the events A = {d, e, a}, B = {c, e, a}, C = {c, d, a}. Show that
(a) What is the probability that your bridge partner has exactly two aces, given that she has at least one ace?(b) What is the probability that your bridge partner has exactly two aces, given that
Prove that if A and B are independent so are(a) A and ˜B.(b) ˜ A and ˜B.
You are given two urns and fifty balls. Half of the balls are white and half are black. You are asked to distribute the balls in the urns with no restriction placed on the number of either type in an
(Johnsonbough8) A coin with probability p for heads is tossed n times. Let E be the event “a head is obtained on the first toss and Fk the event ‘exactly k heads are obtained.” For which pairs
In London, half of the days have some rain. The weather forecaster is correct 2/3 of the time, i.e., the probability that it rains, given that she has predicted rain, and the probability that it does
A student is applying to Harvard and Dartmouth. He estimates that he has a probability of .5 of being accepted at Dartmouth and .3 of being accepted at Harvard. He further estimates the probability
Let A1, A2, and A3 be events, and let Bi represent either Ai or its complement Ai. Then there are eight possible choices for the triple (B1, B2, B3). Prove that the events A1, A2, A3 are independent
A box has numbers from 1 to 10. A number is drawn at random. Let X1 be the number drawn. This number is replaced, and the ten numbers mixed. A second number X2 is drawn. Find the distributions of X1
Given that P(X = a) = r, P (max(X, Y) = a) = s, and P (min(X, Y) = a) = t, show that you can determine u = P(Y = a) in terms of r, s, and t.
Assume that the random variables X and Y have the joint distribution given in Table 4.6.(br) (a) What is P(X ≥ 1 and Y ≤ 0)?(br) (b) What is the conditional probability that Y ≤ 0
In the previous problem, assume that p = 1 − p.(a) Show that under either service convention, the first player will win more often than the second player if and only if p > .5.(b) In
Let X1 and X2 be independent random variables and let Y1 = Ф1(X1) and Y2 = Ф2(X2).(a) Show that(b) Using (a), show that P (Y1 = r, Y2 = s) = P (Y1 = r) P (Y2 = s) so that Y1 and Y2 are
You are given two urns each containing two biased coins. The coins in urn I come up heads with probability p1, and the coins in urn II come up heads with probability p2 ≠ p1. You are given a
Prove that if P (A\C) ≥ P (B|C) and P (A\ C) ≥ P (B|C), then P (A) ≥ P (B).
George Wolford has suggested the following variation on the Linda problem (see Exercise 1.2.25). The registrar is carrying John and Mary’s registration cards and drops them in a puddle. When he
Let Ri be the event that the ith player in a poker game has a royal flush. Show that a royal flush (A, K, Q, J, 10 of one suit) attracts another royal flush, that is P (R2|R1) > P (R2). Show that a
Prove that A neither attracts nor repels B if and only if A and B are independent.
Prove that if A attracts B, then A repels ˜B .
Prove that if B1, B2, . . . , Bn are mutually disjoint and collectively exhaustive, and if A attracts some Bi, then A must repel some Bj.
Pick a point x at random (with uniform density) in the interval [0, 1]. Find the probability that x > 1/2, given that (a) x > 1/4. (b) x < 3/4. (c) |x − 1/2| < 1/4. (d) x2 − x + 2/9 <
The Acme Super light bulb is known to have a useful life described by the density function f (t) = .01e−.01t, Where time t is measured in hours (a) Find the failure rate of this bulb (see
Suppose you choose two numbers x and y, independently at random from the interval [0, 1]. Given that their sum lies in the interval [0, 1], find the probability that (a) |x − y| < 1. (b) xy <
Let x and y be chosen at random from the interval [0, 1]. Show that the events x > 1/3 and y > 2/3 are independent events.
A coin has an unknown bias p that is assumed to be uniformly distributed between 0 and 1. The coin is tossed n times and heads turns up j times and tails turns up k times. We have seen that the
In the spinner problem (see Example 2.1) divide the unit circumference into three arcs of length 1/2, 1/3, and 1/6. Write a program to simulate the spinner experiment 1000 times and print out what
Alter the program MonteCarlo to estimate the area of the circle of radius 1/2 with center at (1/2, 1/2) inside the unit square by choosing 1000 points at random. Compare your results with the true
Alter the program MonteCarlo to estimate the area under the graph of y = 1/(x + 1) in the unit square in the same way as in Exercise 4. Calculate the true value of this area and use your simulation
In the preceding exercise, it is natural to ask “How do we get the information that the given hand has an ace?” Gridgeman considers two different ways that we might get this information. (Again,
Using the notation introduced in Example 4.29, letShow that there is exactly one value of x such that if your envelope contains x, then you should switch.
Show that, if X and Y are random variables taking on only two values each, and if E(XY ) = E(X)E(Y ), then X and Y are independent.
If the first roll in a game of craps is neither a natural nor craps, the player can make an additional bet, equal to his original one, that he will make his point before a seven turns up. If his
Let X be the first time that a failure occurs in an infinite sequence of Bernoulli trials with probability p for success. Let pk = P(X = k) for k = 1, 2, . . . . Show that pk = pk−1q where q = 1
A multiple choice exam is given. A problem has four possible answers, and exactly one answer is correct. The student is allowed to choose a subset of the four possible answers as his answer. If his
It has been said 12 that a Dr. B. Muriel Bristol declined a cup of tea stating that she preferred a cup into which milk had been poured first. The famous statistician R. A. Fisher carried out a test
In the casino game of blackjack the dealer is dealt two cards, one face up and one face down, and each player is dealt two cards, both face down. If the dealer is showing an ace the player can look
(Feller14) A large number, N, of people are subjected to a blood test. This can be administered in two ways: (1) Each person can be tested separately, in this case N test are required, (2) the blood
The following related discrete problem also gives a good clue for the answer to Exercise 32. Randomly select with replacement t1, t2, . . . , tr from the set (1/n, 2/n, . . . , n/n). Let X be the
A coin is tossed until the first time a head turns up. If this occurs on the nth toss and n is odd you win 2n/n, but if n is even then you lose 2n/n. Then if your expected winnings exist they are
(from Propp18) In the previous problem, let P be the probability that at the present time, each book is in its proper place, i.e., book i is ith from the top. Find a formula for P in terms of the
In a certain manufacturing process, the (Fahrenheit) temperature never varies by more than 2o from 62o. The temperature is, in fact, a random variable F with distribution(a) Find E (F) and V (F).(b)
A number is chosen at random from the integers 1, 2, 3, . . . , n. Let X be the number chosen. Show that E(X) = (n+1)/2 and V (X) = (n−1)(n+1)/12. Hint: The following identity may be useful:
Peter and Paul play Heads or Tails (see Example 1.4). Let Wn be Peter’s winnings after n matches. Prove that E(Wn) = 0 and V (Wn) = n.
Suppose that n people have their hats returned at random. Let Xi = 1 if the ith person gets his or her own hat back and 0 otherwise. Let Sn = Σn i=1 Xi. Then Sn is the total number of people
Let Sn be the number of successes in n independent trials. Use the program BinomialProbabilities (Section 3.2) to compute, for given n, p, and j, the probability P (−j√npq < Sn −
Let X be a random variable taking on values a1, a2, . . . , pr with probabilities p1, p2, . . . , pr and with E(X) = μ. Define the spread of X as follows:This, like the standard deviation, is a
Let X be a random variable with E(X) = μ and V (X) = σ2. Show that the function f(x) defined by
If X and Y are any two random variables, then the covariance of X and Y is defined by Cov(X, Y) = E ((X −E(X))(Y −E(Y ))). That Cov(X, X) = V (X). Show that, if X and Y are independent, then
(Lamperti20) An urn contains exactly 5000 balls, of which an unknown number X are white and the rest red, where X is a random variable with a probability distribution on the integers 0, 1, 2, . . . ,
Referring to Exercise 6.1.30, find the variance for the number of boxes of Wheaties bought before getting half of the players’ pictures and the variance for the number of additional boxes needed to
Let X be a random variable with range [−1, 1] and let fX (x) be the density function of X. Find μ(X) and 2(X) if, for |x| < 1, (a) fX (x) = 1/2. (b) fX (x) = |x|. (c) fX (x) = 1
Let X be a random variable with density function fX. Show, using elementary calculus, that the function Ф(a) = E((X − a)2) takes its minimum value when a = μ(X), and in that case
Let X, Y, and Z be independent random variables, each with mean μ and variance 2. (a) Find the expected value and variance of S = X + Y + Z. (b) Find the expected value and variance of A =
The Pilsdorff Beer Company runs a fleet of trucks along the 100 mile road from Hangtown to Dry Gulch. The trucks are old, and are apt to break down at any point along the road with equal probability.
Let X be a random variable that takes on nonnegative values and has distribution function F(x). Show thatHint: Integrate by parts. Illustrate this result by calculating E(X) by this method if X has
Let X be a random variable distributed uniformly over [0, 20]. Define a new random variable Y by Y = [X] (the greatest integer in X). Find the expected value of Y. Do the same for Z = [X + .5].
Let X and Y be random variables. The covariance Cov(X, Y) is defined by (see Exercise 6.2.23) cov (X, Y) = E((X − μ(X))(Y − μ(Y))) . (a) Show that cov(X, Y) = E (XY) −
Let X and Y be independent random variables with uniform densities in [0, 1]. Let Z = X + Y and W = X − Y. Find (a) p (X, Y ) (see Exercise 18). (b) p (X,Z). (c) p (Y,W). (d) p (Z,W).
For correlated random variables X and Y it is natural to ask for the expected value for X given Y . For example, Galton calculated the expected value of the height of a son given the height of the
A game is played as follows: A random number X is chosen uniformly from [0, 1]. Then a sequence Y1, Y2, . . . of random numbers is chosen independently and uniformly from [0, 1]. The game ends the
Consider the following two experiments: the first has outcome X taking on the values 0, 1, and 2 with equal probabilities; the second results in an (independent) outcome Y taking on the value 3 with
(a) A die is rolled three times with outcomes X1, X2, and X3. Let Y3 be the maximum of the values obtained. Show thatUse this to find the distribution of Y3. Does Y3 have a bell-shaped
Prove that you cannot load two dice in such a way that the probabilities for any sum from 2 to 12 are the same. (Be sure to consider the case where one or more sides turn up with probability zero.)
Suppose again that Z = X + Y. Find fZ if
Suppose that X and Y are independent and Z = X + Y . Find fZ if
Suppose that R2 = X2 + Y 2. Find fR2 and fR if
Particles are subject to collisions that cause them to split into two parts with each part a fraction of the parent. Suppose that this fraction is uniformly distributed between 0 and 1. Following a
Let X1, X2, . . . , Xn be a sequence of independent random variables, all having a common density function fX. Let A = Sn/n be their average. Find fA if (a) fX(x) = (1/√2π) e−x2/2
Find the maximum possible value for p(1 − p) if 0
In Exercise 6.2.15, you showed that, for the hat check problem, the number Sn of people who get their own hats back has E (Sn) = V (Sn) = 1. Using Chebyshev’s Inequality, show that P(Sn ≥ 11)
We have two coins: one is a fair coin and the other is a coin that produces heads with probability 3/4. One of the two coins is picked at random, and this coin is tossed n times. Let Sn be the number
We have proved a theorem often called the Weak Law of Large Numbers.€ Most people€™s intuition and our computer simulations suggest that, if we toss a coin a sequence of times,
Let us toss a biased coin that comes up heads with probability p and assume the validity of the Strong Law of Large Numbers as described in Exercise 15. Then, with probability 1,
Write a program to toss a coin 10,000 times. Let Sn be the number of heads in the first n tosses. Have your program print out, after every 1000 tosses, Sn − n/2. On the basis of this
Let X be the random variable of Exercise 2. (a) Calculate the function f(x) = P (|X − 10| ≥ x). (b) Now graph the function f(x), and on the same axes, graph the Chebyshev function g(x)
Show that, if X ≥0, then P(X ≥ a) ≤E(X)/a.
In each of the following situations, state whether it is a correctly stated hypothesis testing problem and why (a) H0: μ = 25, H1μ != 25 (b) H0: σ > 10, H1: σ = 10 (c) H0: x =
A textile fiber manufacturer is investigating a new drapery yarn, which the company claims has a mean thread elongation of 12 kilograms with a standard deviation of 0.5 kilograms. The company wishes
Repeat Exercise 9-2 using a sample size of n = 16 and the same critical region.
In Exercise 9-2, find the boundary of the critical region if the type I error probability is specified to be.
In Exercise 9-2, find the boundary of the critical region if the type I error probability is specified to be 0.05.
The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100 and the standard deviation is 2. We wish to test versus with a sample
Repeat Exercise 9-6 using a sample size of and the same acceptance region.
A consumer products company is formulating a new shampoo and is interested in foam height (in millimeters). Foam height is approximately normally distributed and has a standard deviation of 20
In Exercise 9-8, suppose that the sample data result in millimeters.(a) What conclusion would you reach?(b) How “unusual” is the sample value millimeters if the true mean is really 175
Repeat Exercise 9-8 assuming that the sample size is n = 16 and the boundary of the critical region is the same.
Consider Exercise 9-8, and suppose that the sample size is increased to n = 16.(a) Where would the boundary of the critical region be placed if the type I error probability were to remain equal to
A manufacturer is interested in the output voltage of a power supply used in a PC. Output voltage is assumed to be normally distributed, with standard deviation 0.25 Volts, and the manufacturer
Rework Exercise 9-12 when the sample size is 16 and the boundaries of the acceptance region do not change.
Rework Exercise 9-12 when the sample size is 16 and the boundaries of the acceptance region do not change. Discuss.
If we plot the probability of accepting H0: μ = μ0 versus various values of μ and connect the points with a smooth curve, we obtain the operating characteristic curve (or the OC curve)
Convert the OC curve in Exercise 9-15 into a plot of the power function of the test.
A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels year-round to reduce air pollution. If more than 400 voters respond positively, we will conclude
The proportion of residents in Phoenix favoring the building of toll roads to complete the freeway system is believed to be p = 0.3. If a random sample of 10 residents shows that 1 or fewer favor
The proportion of adults living in Tempe, Arizona, who are college graduates is estimated to be p = 0.4. To test this hypothesis, a random sample of 15 Tempe adults is selected. If the number of
The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than 100°F. Past experience has indicated that the standard deviation of temperature is 2°F.
Reconsider the chemical process yield data from Exercise 8-9. Recall that = 3, yield is normally distributed and that n = 5 observations on yield are 91.6%, 88.75%, 90.8%, 89.95%, and 91.3%. Use a =
A manufacturer produces crankshafts for an automobile engine. The wear of the crankshaft after 100,000 miles (0.0001 inch) is of interest because it is likely to have an impact on warranty claims. A
A melting point test of n = 10 samples of a binder used in manufacturing a rocket propellant resulted in Assume that melting point is normally distributed with . (a) Test H0: μ = 155 versus H0:
The life in hours of a battery is known to be approximately normally distributed, with standard deviation σ = 1.25 hours. A random sample of 10 batteries has a mean life of hours. (a) Is there

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