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nonparametric statistical inference

Probability And Statistical Inference 3rd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions

Let X and Y have the joint distribution P{X = x, Y = y} = cλx+y/(x!y!) for x = 0, 1, ...,y = 0, 1, ..., and λ > 0. (i) Findc. (ii) Find the marginal distribution of X. (iii) Are X and Y independent?
A box contains three coconut candies, five hazelnut chocolates, and two peanut butter chocolates. A sample of four sweets is chosen from the box. Let X, Y, and Z be the number of coconut candies, hazelnut chocolates, and peanut butter chocolates in the sample, respectively. (i) Find the joint
An urn contains five balls, two of them red and three green. Three balls are drawn without replacement. Let X and Y denote the number of red (X) and green (Y ) balls drawn. (i) Find the joint distribution of (X, Y ). (ii) Find the marginal distributions of X and Y . (iii) Are X and Y independent?
Two cards are drawn at random from the ordinary deck of cards. Let X be the number of aces and let Y be the number of hearts obtained. (i) Find the joint probability function of (X, Y ). (ii) Find the marginal distribution of X and Y . (iii) Are X and Y independent?
The joint probability function of variables X and Y is f(x, y) = c|x − y|, x = 0, 1, 2, 3, and y = 0, 1, 2. Find: (i)c. (ii) P(X = Y ). (iii) P(X>Y ).(iv) P(|X − 1| ≤ 1). (v) P(X + Y ≤ 3).
Variables X and Y have the joint density f(x, y)=1/y for 0 2X).
Let the joint density of random variables X, Y be f(x, y) = cx3y2 for 0 ≤ x ≤ 1, x2 ≤ y ≤ 1 and f(x, y)=0, otherwise. Find P(X
Assume that (X, Y ) have the joint density f(x, y) = cxy2 for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and f(x, y)=0, otherwise. Find: (i)c. (ii) P{X2 ≤ Y ≤ X}. (iii) The cdf of (X, Y ).
Assume that X, Y have density f(x, y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, and f(x, y)=0, otherwise. Find P{Y ≤ √3 X}.
The joint density of X and Y is f(x, y) = y2(xy3 + 1) on the rectangle 0 ≤ x ≤ k, 0 ≤ y ≤ 1. Find: (i) k. (ii) P(X ≤ Y ).
Let the joint cdf of random variables X, Y be F(x, y) = (1/48)xy(x + 2y) for 0 ≤ x ≤ 2, 0 ≤ y ≤ 3. Find the density f(x, y).
A regular die is tossed twice. Let X be the number of times that 1 came up, and Y be the number of times 2 came up. Find: (i) The joint distribution of X and Y .(ii) The correlation coefficient between events {X = 1} and {Y = 2}. [Hint: See formula (4.17) in Definition 4.5.2.]
A regular die is tossed twice. Find: (i) The joint distribution of variables X = the total of both outcomes and Y = the best of the two outcomes. (ii) P(X ≤ 8, Y ≤ 5); P(X = 9, Y ≤ 2), P(4 ≤ X ≤ 7, 1 ≤ Y ≤ 3).(iii) P(Y = 3|X = 4), P(Y < 6|X = 7), P(4 < Y ≤ 6|X ≤ 8).
A cancer specialist claims that the hazard function of random variable Tc = “age at death due to cancer” is a bounded function which for large t has the form hc(t) = k/t2(where t is the age in years and k is some constant).Assume that h(t) is the hazard of the time of death due to other reasons
The series system is built in such a way that it operates only when all its three components operate (so it fails when at least one component fails). Assuming that the lifetime of each component has EXP (1) distribution and that the components operate independently, find the distribution and hazard
Let two distributions with densities f1 and f2 have constant hazard rates λ1 and λ2 =aλ1, respectively. Show that S1(t) = S2(t)a, if S1(t) and S2(t) are respective survival functions.
Assume that the fuel pumps in a certain make of cars have lifetimes with a Weibull hazard rate 2/√3 t (t measured in years). Find the probability that a fuel pump is still working after 5 months.
Find the cdf and density of Weibull distribution with the hazard function (5.45).
Let X be a random variable with density f(x) =⎧⎨⎩0 x < 0 0.5 0 ≤ x ≤ 1 qe−αx x > 1.(i) Find q if α is known. (ii) Find hazard h(t) and survival function S(t).
Find the density and survival function of the distribution with hazard rate h(t) = a +bt for t > 0,a> 0,b> 0.
Find the hazard function of the U[0, 1] distribution. Explain why h(t) is unbounded.
The speed of a molecule of gas at equilibrium is a random variable X with density f(x) = kx2e−bx2 for x > 0 0 otherwise, where k is a normalizing constant and b depends on the temperature of the gas and the mass of the molecule. Find the probability density of the kinetic energy E = mX2/2 of the
Suppose that the measured radius R is a random variable with density fR(x) = 12x2(1 − x) for 0 ≤ x ≤ 1 0 otherwise.In a circle with radius R, find the cdf of: (i) The diameter. (ii) The area.
Let X have density f(x) = 2(1 − x), 0 ≤ x ≤ 1. Find the density of: (i) Y =X(1 − X). (ii) W = max(X, 1 − X).
The duration (in days) of the hospital stay following a certain treatment is a random variable Y =4+ X, where X has a density f(x) = 32/(x + 4)3 for x > 0. Find:(i) The density of Y . (ii) The probability that a randomly selected patient will stay in the hospital for more than 10 days following the
Let X have U[−1, 1] distribution. Find the distribution of: (i) Y = |X|.(ii) Z = 2|X| − 1. (iii) Solve (i) and (ii) if the distribution of variable X is U[−1, 2].
Random variable X has density fX(x) = Cx4 for −2 ≤ x ≤ 1 and 0 otherwise. Find the density of Y = X2 (follow Example 5.26).
Find the density of Y = X(1 + X) if X has U[0, 1] distribution.
Assume that X has the standard normal distribution. Find the density of: (i) Y =X3. (ii) Y = (X − 1)3. (iii) Y = eX.
Let X have EXP(λ) distribution, and let Y = √X. Find: (i) The cdf and density of Y .(ii) The lower quartile of Y .
Let X be U[0, 1]. Find ϕ such that Y = ϕ(X) has EXP(λ) distribution.
Let X have a continuous distribution with cdf FX and density fX, such that FX(0) =0. Find the density of variables: (i) √X. (ii) log X. (iii) 1/X. (iv) eX.
Let X have the Poisson distribution with parameter λ, and let Y = 2X. Find the distribution of Y .
If X is the result of tossing a balanced die, find the distribution of: (i) Y = (X − 2)2.(ii) Z = |X − 2.5|.
An oscillator sends the wave X(t) = A cos(2πt), where A = 1 or 2 with equal probabilities. We observe the value of X(t) at the point chosen at random from the U[n,n + 1] distribution for some n. Find: (i) P(X(t) ≤ 1). (ii) P(|X(t)| > 3/2).(iii) P(X(t) > 0).
Light bulbs of a certain type, manufactured by companies A, B, and C, have lifetime distributions EXP(1), EXP(2), and EXP(3), respectively. If bulbs are mixed in equal proportion in a box, find the probability that a randomly selected bulb will work for at least 1 year.
Let random variable X with the cdf F be uniformly distributed over the union of intervals (0,a) and (a + 2,b). Assuming that F(4) = 0.2 and F(a + 1) = 0.25, find:(i) a andb. (ii) F(8.39). (iii) P(3.01 ≤ X ≤ 9.14). (iv) The probability that among 10 values independently selected from this
Let X have the density f(x) = Cx for 0 ≤ x ≤ 1,f(x) = C(2 − x)/2 for 1 < x ≤ 2, and f(x)=0 otherwise. Find C and F(x). Compute the following probabilities and show them on the graphs of f(x) and F(x): (i) P(X ≥ 3/2). (ii) P(|X − 1| ≤ 1/2).(Hint: The problem can be solved without
Let Xn be the difference (possibly negative) between the number of heads and the number of tails in n tosses of a coin. Find: (i) The distribution of X4. (ii) The cdf of X4 at point x = −0.6. (iii) The probability that Xn is positive given that it is nonnegative for (a) n = 4 and (b) n = 5.
Let X have EXP(λ) distribution. Show that for s,t > 0 the following memoryless property holds: P{X>s + t|X>s} = P{X>t}.
Let X have EXP(λ) distribution, and let Y and Z be defined as in Problem 5.3.4. (i) Find P(Y = Z). (ii) Show that P(Y = k|Z = k + 1) = P(Y = Z) for all k = 0, 1, .... (iii) Find P(Z = k + 1|Y = k) for k = 0, 1, ....
Let X have EXP(1) distribution. Moreover, let Y = [X] be the integer part of X, and let Z be the integer nearest to X. Find: (i) The distributions of Y and Z.(ii) P(Y = Z). (iii) P(Y = 3|Z = 4). (iv) P(Z = 4|Y = 3). (v) P(Y = 4|Z = 3).(vi) P(Z = 3|Y = 4).
Let X have the density f(x) = Ce−0.4|x|, −∞ −2). (ii) P(|X +0.5| < 1).
You have five coins in your pocket: two pennies, two nickels, and a dime. Three coins are drawn at random. Let X be the total amount drawn (in cents). Find: (i) The distribution of X. (ii) P(X ≤ 10|X ≤ 15). (iii) The probabilities that two pennies are drawn, if it is known that X ≤ 11.
A die is biased in such a way that the probability of obtaining k dots (k = 1, ..., 6)is proportional to k2. Which number of dots is more likely: odd or even?
A coin of diameter d is dropped on a floor covered with square tiles with side length D>d. Let X be the number of tiles which intersect with the coin. (i) Find the distribution of X. (ii) Determine the median of X as a function of D and d.
A point is chosen at random from a square with sidea. Let X be the distance from the selected point to the nearest corner of the square. Find and graph FX(x).
Determine the medians and lower and upper quartiles for random variables with the following cdf’s:(i) FX(x) =⎧⎨⎩0 for x < 0 kx3 for 0 ≤ x ≤ 1/√3 k, 1 for x > 1/√3 k, k > 0(ii) FX(x) =0 for x < 0 1 − αe−x for x ≥ 0. (consider all possible α)
Let X be a random variable with cdf given by FX(x)=0 for x < 0 and FX(x)=1 −0.3e−λx for x ≥ 0. Determine: (i) P(X = 0). (ii) λ if P(X ≤ 3) = 3/4. (iii) P(|X| ≤ 5)using results of (ii)
Figure 5.3 shows the cdf of a random variable X. Find: (i) P(X = −2), P(X = 0).(ii) P(X ≤ 3), P(X < 3), P(X < 0.13). (iii) P(X > 2), P(X > 2.79). (iv) P(−1
In the statements below, F and G stand for cdf’s of random variables X and Y, respectively. Classify each of the statements below as true or false: (i) If X is always strictly positive, then F(t) is strictly positive for all t. (ii) If F(37) = F(45), then P(40
In Problem 4.7.9, it was assumed that every man had one son. Assume now that the probability that a man has a son is r. Define a Markov chain with four states, where the first three states are as in Problem 4.7.9, and the fourth state is entered when a man has no son. This state cannot be left (it
Assume that a man’s occupation can be classified as professional, skilled laborer, or unskilled laborer. Assume that of the sons of professional men, a percent are professional, the rest being equally likely to be skilled laborers or unskilled laborers.In the case of sons of skilled laborers, b
Find all two-step transition probabilities for: (i) The Markov chain described in Problem 4.7.4. (ii) The dog flea model of Problem 4.7.6.
Consider a specific kind of part needed for the operation of a certain machine (e.g., the water pump of a car). When the part breaks down, it is replaced by a new one.The probability that a new part will last for exactly n days is rn,n = 1, 2, ... . Let the state of the system be defined as the age
(Dog Fleas, or the Ehrenfest Model of Diffusion) Consider two urns (or dogs), and N balls (or fleas), labeled 1, ...,N, allocated between the urns. At times t = 1, 2, ..., a number 1 through N is chosen at random, and the ball with the selected number is moved to the other urn. Let s(n) be the
A college professor teaches a certain course year after year. He has three favorite questions, and he always uses one of them in the final exam. He never uses the same question twice in a row. If he uses question A in one year, then the next year, he tosses a coin to choose between question B and
Suppose the results of an election in a certain city are found to depend only on the results of the last two elections. Specifically, letting R and D denote Republican and Democratic victories, the state before any election may be RR, RD, DR, DD, the letters signifying, respectively, the outcomes
(i) Modify Example 4.20 by assuming that in each game, the player may win with probability p, lose with probability q, or draw (so that his fortune does not change)with probability r, where p + q + r = 1. Find the transition probability matrix in this case. (ii) Modify Example 4.21 same way as in
After some time spent in a bar, Peter starts to walk home. Suppose that the streets form a rectangular grid. Peter always walks to the nearest corner and then decides on the direction of the next segment of his walk (so that he never changes direction in the middle of the block). Define the
Customers arrive at a service station (a taxi stand, a cable car lift at a skiing resort, etc.) and form a queue. At times t = 1, 2, ... the first m customers (m ≥ 1) in the queue are served (if there are that many). Let Y1,Y2, ... denote the numbers of customers arriving during the time
Generalizing the scheme of Problem 4.6.1, let pt be the probability of choosing size tfor subset St ⊂ {1, ...,N}, 1 ≤ t ≤ N. After choosing t, subset St is selected at random, and all events with indices in St occur while other events do not. (i) Argue that events A1, ...,AN are exchangeable.
A subset S of size t, 1 ≤ t ≤ N, is selected at random from the set {1, ...,N} and event Ai, i = 1, ...,n, is defined as “Element i was among the elements selected.” If S ={i1, ...,it} is chosen, we say that events Ai1 , ...,Ait occur, while the remaining events do not. (i) Show that events
Is it possible to bias a die in such a way that in tossing the die twice, each sum 2, 3, ..., 12 has the same probability?
The French mathematician Jean D’Alembert claimed that in tossing a coin twice, we have only three possible outcomes: “two heads,” “one head,” and “no heads.” This is a legitimate sample space, of course. However, D’Alembert also claimed that each outcome in this space has the same
A machine has three independent components, two that fail with probability p and one that fails with probability 0.5. The machine operates as long as at least two parts work. Find the probability that the machine will fail.
Consider a die in which the probability of a face is proportional to the number of dots on this face. What is the probability that in six independent throws of this die each face appears exactly once?
Three people, A, B, and C, take turns rolling a die. The first one to roll 5 or 6 wins, and the game is ended. Find the probability that A will win.
Find the probability that in repeated tossing of a pair of dice, a sum of 7 will occur before a sum of 8.
A coin with probability p of turning up heads is tossed until it comes up tails. Let X be the number of tosses required. You bet that X will be odd, and your opponent bets that X will be even. For what p is the bet advantageous to you? Is there a p such that the bet is fair?
Two people take turns rolling a die. Peter rolls first, then Paul, then Peter again, and so on. The winner is the first to roll a six. What is the probability that Peter wins?
A coin is tossed six times. Find the probability that the number of heads in the first three trials is the same as the number of heads in the last three trials.
Events A and B are such that 3P(A) = P(B) = p, where 0
Suppose that a point is picked at random from the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.Let A be the event that it falls in the triangle bounded by the lines y = 0,x = 1, and x = y, and let B be the event that it falls into the rectangle with vertices (0, 0),(1, 0),(1, 1/2), and (0, 1/2). Find
The probability that a certain event A occurs at least once in three independent trials exceeds the probability that A occurs twice in two independent trials. Find the possible values of P(A).
Let X be the number on the chip randomly selected from a box containing 12 chips, labeled 1 through 12. Check pairwise independence of events A,B, and C, defined as X is even, X ≥ 7, and X < 4, respectively.
If disjoint events A and B have positive probabilities, check independence of events in the following pairs: ∅ and A, A and B, A and S, A and A ∩ B, ∅ and Ac.
Events A and B are independent, A and C are mutually exclusive, and B and C are independent. Find P(A ∪ B ∪ C) if P(A)=0.5,P(B)=0.25, and P(C)=0.125.
A die is tossed three times, with outcomes X1,X2, and X3. Assuming that all 216 possible outcomes (x1,x2,x3) are equally likely, find following probabilities:(i) P(X1 > X2 = X3). (ii) P(X1 < X2 < X3). (iii) P[max(X1,X2,X3) = 4].(iv) P[min(X1,X2,X3) = 3].
Let A and B be independent events. Find P(A ∩ B) if: (i) P(A ∩ Bc)=1/3 and P(Ac ∩ B)=1/6. (ii) P(A) = kP(B), and either A or B must occur.
Label the following statements as true or false: (i) The target is to be hit at least once.In three independent shots at the target (instead of one shot), you triple the chances of attaining the goal (assume each shot has the same positive chance of hitting the target). (ii) If A and B are
A prisoner is sentenced to life in prison. One day the warden comes to him and offers to toss a fair coin for either getting free or being put to death. After some deliberation the prisoner refuses, on the ground that it is too much risk: He argues that he may escape, or be pardoned, and so on. The
Twenty chips are placed in a box. On one side each chip has either red (10), blue (5), or green (5) color. On the other side each chip has either one or two dots: six red chips have one dot, while four have two dots, exactly two green and three blue chips have two dots. One chip was randomly
Players A and B draw balls in turn, without replacement, from an urn containing three red and four green balls. A draws first. The winner is the person who draws the first red ball. Given that A won, what is the probability that A drew a red ball on the first draw?
We have three dice, each with numbers x = 1, ..., 6, and with probabilities as follows:die 1: p(x)=1/6, die 2: p(x) = (7 − x)/21, die 3: p(x) = x2/91. A die is selected at random, tossed, and the number 4 appears. What is the probability that it is die 2 that was tossed?
Suppose that box A contains four red and five green chips and box B contains six red and three green chips. A chip is chosen at random from box A and placed in box B.Finally, a chip is chosen at random from those now in box B. What is the probability that a green chip was transferred given that a
One box contains six red and three green balls. The second box has six red and four green balls. A box is chosen at random. From this box two balls are selected and found to be green. Find the probability that the pair was drawn from the first box if the draws are: (i) Without replacement. (ii)
An urn originally contains three blue and two green chips. A chip is chosen at random from the urn, returned, and four chips of the opposite color are added to the urn.Then a second chip is drawn. Find the probability that: (i) The second chip is blue.(ii) Both chips are of the same color. (iii)
Two different suppliers, A and B, provide the manufacturer with the same part. All supplies of this part are kept in a large bin. In the past 2% of all parts supplied by A and 4% of parts supplied by B have been defective. Moreover, A supplies three times as many parts as B. Suppose that you reach
Suppose that medical science has developed a test for a certain disease that is 95%accurate, on both those who do and those who do not have the disease. If the incidence rate of this disease in the population is 5%, find the probability that a person: (i) Has the disease when the test is positive.
Recall Example 4.9. Find the probability that the mother is a carrier if: (i) Both father and son are color-blind, and the mother is not. (ii) It is known only that the son is color-blind. (iii) The son is color-blind, but the parents are not.
(Tom Sawyer Problem) You are given a task, say painting a fence. The probability that the task will be completed if k friends are helping you is pk (k = 0, 1, ...). If j friends already helped you, the probability that the (j + 1)st will also help is πj (j =0, 1, ...). On the other hand, if the
Let A and B be two events with P(B) > 0, and let C1,C2, ... be a possible partition of a sample space. Prove or disprove the following formulas:P(A|B) = i P(A|B ∩ Ci)P(Ci), P(A|B) = i P(A|B ∩ Ci)P(B|Ci)P(Ci).
we return the second ball to the urn, and add new balls as described, with the condition that if the second ball is blue, we add one ball of each color. Then we draw the third ball. What is the probability that the third ball is:(i) Blue? (ii) Blue if the first ball was red? (iii) Blue if the
Suppose that in Problem
Suppose that initially the urn contains one red and two green balls. We draw a ball and return it to the urn, adding three red, one green, and two blue balls if a red ball was drawn, and three green and one blue ball if a green ball was drawn. Then a ball is drawn from the urn. Find the probability
An event W occurs with probability 0.4. If A occurs, then the probability of W is 0.6;if A does not occur but B occurs, the probability of W is 0.1. However, if neither A nor B occurs, the probability of W is 0.5. Finally, if A does not occur, the probability of B is 0.3. Find P(A).
An urn contains three red and two green balls. If a red ball is drawn, it is not returned, and one green ball is added to the urn. If a green ball is drawn, it is returned, and two blue balls are added. If a blue ball is drawn, it is simply returned to the urn. Find the probability that in three
Three cards are drawn without replacement from an ordinary deck of cards. Find the probability that: (i) The first heart occurs on the third draw. (ii) There will be more red than black cards drawn. (iii) No two consecutive cards will be of the same value.
A tennis player has the right to two attempts at a serve: If he misses his first serve, he can try again. A serve can be played “fast” or “slow.” If a serve is played fast, the probability that it is good (the ball hits opponent’s court) is A; the same probability for a slow serve is B.

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