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introduction to probability statistics

Introduction To Probability Volume 2 1st Edition Narayanaswamy Balakrishnan, Markos V. Koutras, Konstadinos G. Politis - Solutions

We then define X1 : the sum of the two numbers in the balls drawn, X2 : the absolute difference between the two numbers in the balls drawn.(iii) We throw a die three times and let Z1, Z2, Z3 denote the respective outcomes.We then define(iv) We throw one die and toss three coins and define X1 : the
(One-sided Chebyshev bound) Let X be a random variable such that E(X) = 0 and Var(X) = ????2. Show that, for any a > 0,This result is known as Cantelli’s inequality.(Hint: For b > 0, put W = (X + b)2 and apply Markov’s inequality to W; then examine which value of b minimizes the quantity
Let X be a discrete random variable, and a and t be two positive real numbers. Assuming that the expectation E(eaX) exists, establish the following inequality P(X
For the random variable X, assume that the expectation E(eaX) exists for a given a ∈ ℝ. Prove that, for any real t, the following holds: P(X t) E(eax) eat
For a discrete random variable X with mean ????, assume that the central moment of order 2r, exists for a given integerr 1. M2r=E[(X-)] Show that for any a > 0, we have P(X ua)
Suppose that X has range {−2, 0, 2} and probability functionExamine whether Chebychev’s inequality is exact, i.e. it holds as an equality in this case. 1/8, x=-2, f(x) = 3/4, x = 0, 1/8, x = 2.
A random variable X has the following probability function(i) Find the expected value of X (a) from Definition 4.4, (b) using Proposition 4.13, and verify that the two results agree.(ii) Find the percentage error in the upper bound obtained from Markov’s inequality as an estimate for the
Markov’s inequality (Proposition 4.11) concerns a nonnegative random variable.An extended version of this inequality that applies to any random variable X is the following:for any t > E(IXI) P(|X| t) t
Consider two random variables X and Y with the same range RX = RY = {a1, a2, a3}, and suppose that E(X) = E(Y) and Var(X) = Var(Y).(i) Prove that E(X2) = E(Y2).(ii) Show that the quantities xi = P(X = ai) − P(Y = ai), i = 1, 2, 3, satisfy a homogeneous linear system of equations, for which the
Let X be a discrete random variable and assume that it has finite moments of any order up to a positive integer r, i.e. ????′i ????r = [E(X − ????)r],where ???? = E(X), is given by the formula(Hint: Begin by expanding the power (X − ????)r according to the binomial theorem.) = Mr -- k=0
If for a variable X it is known that E(X − 1) = 3, E[X(X + 2)] = 40, calculate the mean and standard deviation of the random variableWith reference to Example 4.16, concerning Nick’s profit from his investment:(i) Find the standard deviation of his profit.(ii) What is the probability that
Find the variance of the random variable X having a probability function f and range RX in each of the following cases: 2-x-21 (i) f(x)= ,x Rx = {1,2,3}; 4 (ii) f(x) = (iii) f(x)= 4+x-12 42 x-3+1 28 , x = Rx = {1,2,3); ,x Rx = (-3, -2, -1, 0, 1, 2, 3}; 4-x (iv) f(x)= 6 x = Rx = {1,2.3).
The following result is known from calculus:Suppose X is a random variable whose probability function has the form f (x) = cx−2, x ∈ RX = {1, 2, 3,…}.(i) Find the value of the constant c.(ii) Show that the expectation of X does not exist. IM8 x=1 || 917
Suppose that a discrete random variable X has range {−4,−3,−2,−1, 0, 1, 2, 3, 4}and the following probability function(i) Show that the random variable Y = |X| has probability function (ii) Find the expectation for each of the variables X and Y and examine whether |E(X)| = E(|X|). fx(x) = x
We consider a random variable X whose range is the set {1, 2, 3,…} and which has probability functionConsider also another variable Y defined asi) Find the probability function and the distribution function of the variable Y.(ii) Calculate E(Y). f(x) = P(X = x) = = x) = 1 x = 1, 2,... 2
A bookstore buys 15 copies of a book at a price of $20 each and sells them at a price of $30. Suppose that, after a year, the bookstore may return any unsold copies of the book to the publisher at the original price of $20. Let X be the number of copies sold during the year and assume that the
The number of new friends that Jane makes on Facebook during a day is a random variable Y with probability functionObtain the probability function and the expected value for the number of friends that she makes during a weekend, i.e. in a two-day period. 0.15, y=1, 0.25, y = 2, f(y)= 0.30, y = 3,
Find the expected value of the random variable X in each of the three cases below, where f (x) denotes the probability function of X: (i) X 012 2 3 f(x) 0.5 0.1 0.3 0.1 (ii) (iii) x 1 2 34 f(x) 0.2 0.3 0.4 0.1 x 2 4 68 f(x) 0.1 0.2 0.3 0.4
Obtain the value of the constant c and the probability function f(x) = P(X = x), x = 1, 2,... Hint: You may use the formula k=1 k - In(1 - 0), 0 <
Empirical studies have suggested that the number of butterflies in a certain area is a random variable, denoted by X, with probability functionwhere, as usual, [x] denotes the integer part of x and ???? is a parameter such that 0 F(x)= C [x] k=1 -
In each case below, find the value of the constant c so that the corresponding function defines a probability function with the given range of values, RX: C (i) f(x) = x(x+1) (ii) f(x)=c 2*, (iii) f(x)= c.2, Rx = {1,2,3,...}; Rx = {0, 1, 2,... }; Rx = {0,1, 2,...}. Hint: for Part (i), observe that
Let X be a discrete random variable whose range is the set of positive integers and which has probability function(i) Verify that this function satisfies Properties (PF1), (PF2), and (PF3) of Proposition 4.3, so that it is indeed a probability function.(ii) Find the distribution function F(t) of
In the random experiment of throwing two dice, let X be the random variable that represents the sum of the two outcomes. Show that the probability function of X is given by f(x)= x-1 36 12-(x-1) x = 2,3,...,7, x 8,9,..., 12. 36
The probability function of a random variable X is given byFind the distribution function F of X and plot this function. X f(x)= x = 1,2,3,4. 10'
In each of the following cases, find the value of the constant c so that the functions given below can be used as probability functions of a random variable X on the range RX provided for each case: (i) f(x) = c(x+5), Rx = {0,1,2,3,4}; (ii) f(x) = c(2x+1), Rx = {1,2,..., 10};
Explain which of the following functions can be used as probability functions of a random variable X with the corresponding range RX given below: x+3 (i) f(x) = Rx = {1,2,3}; 15 2x-3 (ii) f(x) = Rx = {1,2,3,4); 16 3x-1 (iii) f(x) = Rx = {1,2,...); 4 x (iv) f(x)= Ry = {1, 2, ..., n}; n(n + 1)(2n+1)*
Let X be a random variable with distribution function F and Y be another random variable which is defined aswhere a and b are two real numbers. Write the distribution function FY of Y in terms of F, and b. X, Y = b, if X a, if X > a,
The range of values for a random variable X is the interval RX = (0, 1), while its distribution function is given byFind the distribution function of the random variable Y = X∕(1 + X). 0, t
Let X be a random variable with distribution function F. The real number m that satisfies F(m−) ≤ 0.5 ≤ F(m)is called the median of this distribution function. Assuming that F is known, and that it is continuous for any real t, explain how m can be found from F.Application: The rate of
AUniversity lecturer ends her lecture in a particular course she gives on Wednesdays between 11:00 a.m. and 11:04 a.m. Suppose that the time X (in minutes) between 11:00 a.m. and the end of the lecture has the following distribution functionfor a positive constant c.We assume that the lecture
The distribution function of a random variable X is given byCalculate the following probabilities 0, (1+1) - 2 (1-1)2 2 -8
The orders for CDs (in hundreds of thousands of units) received daily by a factory that produces compact discs are represented by a random variable X with distribution function(i) Find the value of the constant ???? above, assuming that the number of CDs ordered daily is less than 100 000.(ii) Find
Examine which of the functions below can be used as distribution functions of a random variable X. (i) F(t) = = { 1- 1-e-3, t0, t
At the end of each year, a company gives a bonus to all of its employees. The size of this bonus can take one of four possible values (in thousands of dollars): 1, 2, 5, Let X be the size of the bonus given to a randomly chosen employee. The distribution function of X is given by(i) What are the
Jenny is waiting at a bus stop for the bus that takes her to work. It is known that the waiting time (in minutes) for this particular bus, represented by a random variable X, has the following distribution function:(i) Draw a graph of this distribution function. Are there any discontinuity points
A random variable X has a distribution function F given by(i) Explain whether X is a discrete or a continuous random variable and identify the range of its values.(ii) Using the distribution function, calculate each of the following probabilities:P(X ≤ 3), P(X 2), P(2 ≤ X ≤ 4). 0, t
Consider the identity(i) Verify that the derivative, of order n − 1, of the function(ii) By differentiating both sides of (2.17) n − 1 times establish that(iii) Making the change of variable j − n + 1 = k on the right-hand side of the last expression, obtain an alternative proof of
The chess clubs of two schools compete against each other every year. Each school team has n players and on the day of the contest, each member of the first team is drawn to play against a member of the second team. In two consecutive contests between the two teams, each of the two teams has the
Considering the special case of Pascal’s triangle (k = 5)and adding side-by-side the resulting equations for j = 5, 6,…, 10, check that the following identity holds true:Using a similar method, generalize the previous result to show that for any positive integers n and k with k ≤ n, we
(The Mathematics of poetry5) The Indian writer Acharya Hemachandra (c. 1150 AD) studied the rhythms of Sanskrit poetry. Syllables in Sanskrit are either long (L)or short (S). Long syllables have twice the length of short syllables. The question he asked is how many rhythm patterns with a given
In a tennis tournament, 16 female players will compete for the title. In the first round of the tournament, the players will form randomly eight pairs and each pair will play a match. The eight winners of these matches will then form four new pairs. Each of these new pairs will play a match and the
A train has three coaches and when it stops at the first station during its journey, n passengers embark it, for n > 3. Assuming that passengers embark on a particular coach of the train independently of one another, what is the probability that at least one passenger embarks on each coach?(Hint:
In the World Cup of football, there are 32 teams that participate. After a draw is made, these teams are divided into eight groups with four teams each. How many different ways there are to form the eight groups?
When we throw three dice, which one is more likely: that the sum of the three outcomes equals 10 or that it equals 9?(This problem is historically associated with the name of Galileo Galilei(1564–1642), who calculated the probabilities of both these events and showed that they were not equal, as
In Jenny’s shoe rack, there are seven pairs of shoes. She picks six shoes at random.What is the probability that among these(i) there is no pair of shoes?(ii) there is exactly one pair of shoes?(iii) there are exactly two pairs?(iv) there are exactly three pairs?
There are n couples competing in a dancing competition. Suppose there are n prizes available in total. What is the probability that exactly one person from each couple wins a prize?
In a city with n + 1 inhabitants, a person tells a rumor to a second person, who then repeats it to a third person, and so on. At each step, the person who tells the rumor chooses the next recipient at random among the n people available.Find the probability that the rumor will be told r times
Peter tosses a die k times for some 2 ≤ k ≤ 6. Find the probability that the results of this die tosses(i) are all the same;(ii) contain at least two outcomes which are the same.
Let X = {x1, x2,…, xn} and k be a positive integer with k ≤ n. Denote bythe ratio of the number of k-element permutations among the n elements without repetitions to the number of k-element permutations with repetitions.We also definewhich is the corresponding ratio for combinations instead of
A bowl contains n lottery tickets numbered 1, 2,…, n.We select a ticket at random, record the number on it, and put it back in the bowl. The same procedure is followed a total of k times.Find the probability that(i) the number 1 is selected at least once;(ii) both numbers 1 and 2 are selected at
Suppose that n students, say S1, S2,…, Sn, apply for a postgraduate program, and they all receive an invitation to attend an interview. They all arrive in time and they are called upon one after the other.(i) Find the number of different ways for the order they will be called for the
Nick has six mathematics books, five physics books, three chemistry books, seven French books, and two dictionaries. He buys a new bookshelf and his mother places the books at random on the shelf. What is the probability that all books of the same subject are placed together?
Consider the following experiment: we ask a computer to select an integer at random from the set {0, 1, 2,…, 99 999}. What is the probability that the number selected(i) does not contain the digit 7?(ii) contains both digits 5 and 7?(iii) contains, at least once, each of the digits 1, 3, 5, 7?
The coefficients of the quadratic polynomial (a + 1)x2 + bx + c are decided from the outcomes of a fair coin, which is tossed three times. Specifically, if at the first toss the coin lands heads we put a = 1, otherwise we put a = 0. Similarly, b = 1 if at the second toss the coin lands heads (b = 0
In a certain town, the percentage of people with blood type A is approximately the same to that with blood type O. Moreover, the percentage of inhabitants with blood type B is about 1∕5 compared to that of type A and three times as much compared to the percentage of persons with type AB. Estimate
Each student, in a class of 50 students, has to choose two optional courses among the four optional courses offered by their Department, so that there are(4 2)alternative choices (course combinations) for any of them. What is the probability that there is exactly one course combination not chosen
A computer machine selects at random an even integer which is greater than or equal to 10 000 and less than 70 000. What is the probability that the number selected has no two digits which are the same?
A safe box requires a four-digit number to be opened.(i) Find the number n of different numbers that are possible codes for the safe box.(ii) If a burglar chooses at random four-digit numbers trying to open the box, find the probability that he succeeds(a) at the kth attempt (1 ≤ k < n);(b)
A company of n students decides to have lunch at the Student Union on a particular day. Each of the students will go alone as they attend different classes in the morning. However, there are k restaurants at the Student Union and they have not specified at which one they are supposed to meet.(i)
We ask 15 people about the day of the week they were born. What is the probability that no one was born on a Sunday?
We want to place n mathematics books and k physics books on a book-shelf.(i) How many ways are there for placing the books on the shelf?(ii) If k ≤ n + 1, find the probability that no two physics books are put next to one another.
A sprinter is about to run a 100 m race on lane 3 of a track that has 8 lanes. If two other runners from the same country take part in this race, what is the probability that neither of them will run next to him?
When we throw simultaneously three identical dice, the number of distinguishable outcomes is (a) (9) (b) (9) (9) (9) (d) 63 (e) 62
The number of different domino tiles that can be formed if the number on each side of a tile is chosen from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is(a) 45 (b) 55 (c) 63 (d) 36 (e) 72
The value of the sum is + (2) k=1 (a) n(n - 1)2"-2 (d) n2"-2 +1 (b) n(n+1)2"-2 (e) n2"-1 (c) n2n-2
The value of the sum 5 10 (1) (5k) k=0 is 16 (a) (15) b)(19)+(5) c) ( (9) c) (19) (9) (d) 25 (e) 216
In the parking area of a supermarket, there are 40 parking places, numbered 1–40. At a particular instant, 15 places are occupied. Assuming that all places are equally likely to be occupied, the probability that exactly one of the parking places numbered 1 and 2 is occupied equals 40 13 40 (19)
There are six red balls and three black balls in a box. All nine balls are identical except their color. Nick claims that he can pull out six balls (without replacement)and that all six will be red. Assuming that he has no magical skills and he chooses at random, let x be the probability that he
During a hand of poker, a player receives 5 cards out of 52 cards from an ordinary deck. The probability that, at a given hand, he receives at least one ace is (a) 13 (b) (c) (+)(43) 52 10 (d) (5)(47) (52) 48 5 (e) 1 52
Marie, as she opens her email, finds out that she has five new messages, of which three are spam messages. The probability that the two nonspam messages were received one after the other, without any spam messages between them, is (a) (b) (c) 215 (d) 30 10 (e) 30
A mathematics teacher has four ties and each day he selects at random one of them to wear to work. In a period of three days, the probability that he does not wear the same tie more than once equals 34 (a) 12 (b) (c) + 318 @ 36 (e)
Ariadne and Athena are schoolmates. The probability that they will celebrate their next birthday on the same day of the week is 67 {(q) 27 (c) (d)
Suppose that there are five red balls and seven black balls in a box. If we select three balls at random and without replacement, the probability that none of them is black equals (b) (c) 3 (5)+(6) (12) (3) (d) (e) +(6)
For nonnegative integers n and k with k ≤ n, the ratio (n)k (k) equals (a) k! (b) (c) (d) (e) n
If we select 2 cards without replacement from an usual pack of 52 cards, the probability that neither of them is an ace is (a) 48 48!2! 52! 48!4! 482 (b) (c) 52 2 (52) (d) 48!4! 52! 522
If we select 3 cards without replacement from an usual pack of 52 cards, the number of possible outcomes is 49 49! (a) (b) (c) 52 523 52 3 (d) 3! (e) 3!
A high-school offers four language courses: French, Spanish, German, and Italian.Each student has to select exactly two of these courses. Assuming that all choices are equally likely, the probability that a student chooses French and Italian is 3 91 (b) (c) (d) 310 23
In three throws of a die, the probability that no two outcomes are the same equals(a) 2 9(b) 20 63 (c) 1 3(d) 5 9(e) 4 9
In two throws of a die, the probability that the sum of the two outcomes equals three is(a) 3 6(b) 1 6(c) 1 36(d) 1 18(e) 4 36
When we throw a die three times, the probability that no six appears is(a) 13 63 (b) 13 53 (c) 53 63 (d) 1 − 13 63 (e) 1 − 53 63
For nonnegative integers m, n, and r with r ≤ n + m, the sumequals 2m+n. m n (7) (~k) k=0 k r
For nonnegative integers n and k with k ≤ n, the sumequals 2n. n () k=0
The coefficient of x4 in the binomial expansion of (1 − x)−6 is(9 4).
The coefficient of x2 in the binomial expansion of (x + 3)6 is(6 2 ).
Three friends throw one die each. The probability that the sum of the three outcomes is 17 equals 1∕72.
A bookshelf contains 30 books. Among them, 12 are fiction and the remaining are nonfiction books. If we select five books at random, the probability that there will be exactly four fiction books among them is 4.18 201 5
Let n and r be two integers with r > n ≥ 1. Then both [n r] and(n r ) are equal to zero.
For nonnegative integers n and r with r ≤ n, it is always true that n r [ 7 ] (7) n
For nonnegative integers n and r with r ≤ n, it is always true that n (7)=(27) n-
For any nonnegative integer n, we have (n+1)= n+1.
Five students are ranked in terms of their academic performance. The total number of possible rankings is 5!
We select three digits from one to nine without replacement. The number of all possible outcomes, in order to form a three-digit number, is 93.
We select three digits from one to nine with replacement. The number of all possible outcomes, in order to form a three-digit number, is (9)3.
The number of different 5-member committees, consisting of 3 men and 2 women, that can be formed among 18 men and 12 women is(18 3)(12 2).
Nick rolls a die twice. The probability that the second outcome is higher than the first is 1∕2.
A box contains 50 lottery tickets, numbered 1, 2,…, 50. We select one ticket at random. The probability that the number on it is a multiple of 3 is 1∕3.
When we throw a die, the probability that the outcome is greater than or equal to 3 is 1∕2.
When we toss a coin three times, the sample space with all different outcomes of the three tosses has eight elements.
For large values of a positive integer n, an approximation which is often used for its factorial, n!, is the following, known as Stirling’s formula:Here, the symbol ∼ means that the ratio of the two sides tends to 1 as n → ∞, so that the formula can also be written in the formCalculate the
Let X = {x1, x2,…, xn} be a finite set and k be a positive integer such that k ≤ n.We denotefor the ratio of the number of k-element permutations among the n elements without repetitions to the number of k-element permutations with repetitions.We also define correspondinglyfor the ratio of

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