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statistical techniques in business
Questions and Answers of
Statistical Techniques in Business
Let X be a number selected at random from a set of numbers {51, 52, . . . , 100}. Approximate E(1/X).
One of the numbers 1, 2, . . . , 6 is to be chosen by casting an unbiased die. Let this random experiment be repeated five independent times. Let the random variable X1 be the number of terminations
Show that the moment generating function of the negative binomial distribution is M(t) = pr[1 − (1 − p)et]−r. Find the mean and the variance of this distribution.
Let the mutually independent random variables X1, X2, and X3 be N(0, 1), N(2, 4), and N(−1, 1), respectively. Compute the probability that exactly two of these three variables are less than zero.
Find the uniform distribution of the continuous type on the interval (b, c) that has the same mean and the same variance as those of a chi-square distribution with 8 degrees of freedom. That is, find
Find the union C1 ∪ C2 and the intersection C1 ∩ C2 of the two sets C1 and C2, where(a) C1 = {0, 1, 2, }, C2 = {2, 3, 4}.(b) C1 = {x : 0 < x < 2}, C2 = {x : 1 ≤ x < 3}.(c) C1 = {(x, y)
Let a card be selected from an ordinary deck of playing cards. The outcome c is one of these 52 cards. Let X(c) = 4 if c is an ace, let X(c) = 3 if c is a king, let X(c) = 2 if c is a queen, let X(c)
Let X be a random variable with mean μ and let E[(X − μ)2k] exist. Show, with d > 0, that P(|X − μ| ≥ d) ≤ E[(X − μ)2k]/d2k. This is essentially Chebyshev’s inequality when k = 1.
A positive integer from one to six is to be chosen by casting a die. Thus the elements c of the sample space C are 1, 2, 3, 4, 5, 6. Suppose C1 = {1, 2, 3, 4} and C2 = {3, 4, 5, 6}. If the
Let X equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of X and compute the probability that X is equal to an odd number.
Find the complement Cc of the set C with respect to the space C if(a) C = {x : 0 < x < 1}, C = {x : 5 /8 < x < 1}.(b) C = {(x, y, z) : x2 + y2 + z2 ≤ 1}, C = {(x, y, z) : x2 + y2 + z2 =
For each of the following, find the constant c so that p(x) satisfies the condition of being a pmf of one random variable X.(a) p(x) = c(2/3)x, x = 1, 2, 3, . . . , zero elsewhere.(b) p(x) = cx, x =
Let X be a random variable such that P(X ≤ 0) = 0 and let μ = E(X) exist. Show that P(X ≥ 2μ) ≤ 1/2.
A random experiment consists of drawing a card from an ordinary deck of 52 playing cards. Let the probability set function P assign a probability of 1/52 to each of the 52 possible outcomes. Let C1
Assume that P(C1 ∩ C2 ∩ C3) > 0. Prove thatP(C1 ∩ C2 ∩ C3 ∩ C4) = P(C1)P(C2|C1)P(C3|C1 ∩ C2)P(C4|C1 ∩ C2 ∩ C3).
Suppose we are playing draw poker. We are dealt (from a well-shuffled deck) five cards, which contain four spades and another card of a different suit. We decide to discard the card of a different
For each of the following distributions, compute P(μ − 2σ(a) f(x) = 6x(1 − x), 0 < x < 1, zero elsewhere.(b) p(x) = (1/2 )x, x = 1, 2, 3, . . . , zero elsewhere.
List all possible arrangements of the four letters m, a, r, and y. Let C1 be the collection of the arrangements in which y is in the last position. Let C2 be the collection of the arrangements in
Let pX(x) = x/15, x = 1, 2, 3, 4, 5, zero elsewhere, be the pmf of X. Find P(X = 1 or 2), P(1/2 < X < 5/2), and P(1 ≤ X ≤ 2).
If X is a random variable such that E(X) = 3 and E(X2) = 13, use Chebyshev’s inequality to determine a lower bound for the probability P(−2 < X <8).
Suppose that p(x) = 1/5, x = 1, 2, 3, 4, 5, zero elsewhere, is the pmf of the discrete-type random variable X. Compute E(X) and E(X2). Use these two results to find E[(X + 2)2] by writing (X + 2)2 =
A coin is to be tossed as many times as necessary to turn up one head. Thus the elements c of the sample space C are H, TH, TTH, TTTH, and so forth. Let the probability set function P assign to these
If the sample space is C = C1 ∪ C2 and if P(C1) = 0.8 and P(C2) = 0.5, find P(C1 ∩ C2).
A hand of 13 cards is to be dealt at random and without replacement from an ordinary deck of playing cards. Find the conditional probability that there are at least three kings in the hand given that
Let the random variable X have mean μ, standard deviation σ, and mgf M(t), −h < t < h. Show thatand E (*=*)=0., 5 [(**)*] =₁. (X− E 1,
Let a random variable X of the continuous type have a pdf f(x) whose graph is symmetric with respect to x = c. If the mean value of X exists, show that E(X) = c.
By the use of Venn diagrams, in which the space C is the set of points enclosed by a rectangle containing the circles C1, C2, and C3, compare the following sets. These laws are called the
Let us select five cards at random and without replacement from an ordinary deck of playing cards.(a) Find the pmf of X, the number of hearts in the five cards.(b) Determine P(X ≤ 1).
Let the pmf p(x) be positive at x = −1, 0, 1 and zero elsewhere.(a) If p(0) = 1/4 , find E(X2).(b) If p(0) = 1/4 and if E(X) = 1/4 , determine p(−1) and p(1).
Let the sample space be C = {c : 0 < c < ∞}. Let C ⊂ C be defined by C = {c : 4 < c < ∞} and take P(C) = ∫C e−x dx. Show that P(C) = 1. Evaluate P(C), P(Cc), and P(C ∪ Cc).
Let X have the pdf f(x) = 3x2, 0 < x < 1, zero elsewhere. Consider a random rectangle whose sides are X and (1−X). Determine the expected value of the area of the rectangle.
Show that the moment generating function of the random variable X having the pdf f(x) = 1/3 , −1 < x < 2, zero elsewhere, is M (t) = { e2t-e- 3t 1 t #0 t = 0.
A drawer contains eight different pairs of socks. If six socks are taken at random and without replacement, compute the probability that there is at least one matching pair among these six socks.
If a sequence of sets C1, C2, C3, . . . is such that Ck ⊂ Ck+1, k = 1, 2, 3, . . . , the sequence is said to be a nondecreasing sequence. Give an example of this kind of sequence of sets.
If C1 and C2 are subsets of the sample space C, show that P(C₁ C₂) ≤ P(C₁) ≤ P(C₁ UC₂) ≤ P(C₁) + P(C₂).
Let the probability set function of the random variable X be PX(D) = ∫D f(x) dx, where f(x) = 2x/9, for x ∈ D = {x : 0 < x < 3}. Define the events D1 = {x : 0 < x < 1} and D2 = {x : 2
Let X be a positive random variable; i.e., P(X ≤ 0) = 0. Argue that(a) E(1/X) ≥ 1/E(X)(b) E[−log X]≥ −log[E(X)](c) E[log(1/X)] ≥ log[1/E(X)](d) E[X3] ≥ [E(X)]3.
If the sample space is C = {c : −∞ < c < ∞} and if C ⊂ C is a set for which the integral ∫C e−|x| dx exists, show that this set function is not a probability set function. What
A bowl contains 10 chips, of which 8 are marked $2 each and 2 are marked $5 each. Let a person choose, at random and without replacement, three chips from this bowl. If the person is to receive the
A pair of dice is cast until either the sum of seven or eight appears.(a) Show that the probability of a seven before an eight is 6/11.(b) Next, this pair of dice is cast until a seven appears twice
If a sequence of sets C1, C2, C3, . . . is such that Ck ⊃ Ck+1, k = 1, 2, 3, . . . , the sequence is said to be a nonincreasing sequence. Give an example of this kind of sequence of sets.
Let X be a random variable of the continuous type that has pdf f(x). If m is the unique median of the distribution of X and b is a real constant, show thatprovided that the expectations exist. For
Given the cdfsketch the graph of F(x) and then compute: (a) P(−1/2 < X ≤ 1/2)(b) P(X = 0)(c) P(X = 1)(d) P(2 < X ≤ 3). F(x)= = x < -1 0 2+2 +² -1
Let the space of the random variable X be D = {x : 0 < x < 1}. If D1 = {x : 0 < x < 1/2} and D2 = {x : 1/2 ≤ x < 1}, find PX(D2) if PX(D1) = 1/4.
In a certain factory, machines I, II, and III are all producing springs of the same length. Machines I, II, and III produce 1%, 4%, and 2% defective springs, respectively. Of the total production of
Let X be a random variable such that E[(X −b)2] exists for all real b. Show that E[(X − b)2] is a minimum when b = E(X).
Suppose C1, C2, C3, . . . is a nondecreasing sequence of sets, i.e., Ck ⊂ Ck+1, for k = 1, 2, 3, . . . . Then limk→∞ Ck is defined as the union C1 ∪C2 ∪C3∪· · ·. Find limk→∞ Ck
Let C1, C2, and C3 be three mutually disjoint subsets of the sample space C. Find P[(C1 ∪ C2) ∩ C3] and P(Cc1∪ Cc2).
Bowl I contains six red chips and four blue chips. Five of these 10 chips are selected at random and without replacement and put in bowl II, which was originally empty. One chip is then drawn at
If C1, C2, C3, . . . are sets such that Ck ⊃ Ck+1, k = 1, 2, 3, . . ., limk→∞ Ck is defined as the intersection C1 ∩ C2 ∩ C3 ∩ · · · . Find limk→∞ Ck if(a) Ck = {x : 2 − 1/k <
Consider an urn which contains slips of paper each with one of the numbers 1, 2, . . . , 100 on it. Suppose there are i slips with the number i on it for i = 1, 2, . . . , 100. For example, there are
In an office there are two boxes of computer disks: Box C1 contains seven Verbatim disks and three Control Data disks, and box C2 contains two Verbatim disks and eight Control Data disks. A person is
For every one-dimensional set C, define the function Q(C) = ΣC f(x), where f(x) = (2/3)(1/3)x, x = 0, 1, 2, . . . , zero elsewhere. If C1 = {x : x = 0, 1, 2, 3} and C2 = {x : x = 0, 1, 2, . . .},
Let X denote a random variable for which E[(X − a)2] exists. Give an example of a distribution of a discrete type such that this expectation is zero. Such a distribution is called a degenerate
A bowl contains 16 chips, of which 6 are red, 7 are white, and 3 are blue. If four chips are taken at random and without replacement, find the probability that:(a) Each of the four chips is
A person has purchased 10 of 1000 tickets sold in a certain raffle. To determine the five prize winners, five tickets are to be drawn at random and without replacement. Compute the probability that
If C1 and C2 are independent events, show that the following pairs of events are also independent: (a) C1 and Cc2(b) Cc1 and C2(c) Cc1 and Cc2.
For every one-dimensional set C for which the integral exists, let Q(C) = ∫C f(x) dx, where f(x) = 6x(1 − x), 0 < x < 1, zero elsewhere; otherwise, let Q(C) be undefined. If C1 = {x : 1/4
Let X denote a random variable such that K(t) = E(tX) exists for all real values of t in a certain open interval that includes the point t = 1. Show that K(m)(1) is equal to the mth factorial moment
For each of the following cdfs F(x), find the pdf f(x) [pmf in part (d)], the 25th percentile, and the 60th percentile. Also, sketch the graphs of f(x) and F(x).(a) F(x) = (1+e−x)−1 ,−∞ <
Compute the probability of being dealt at random and without replacement a 13-card bridge hand consisting of: (a) 6 spades, 4 hearts, 2 diamonds, and 1 club(b) 13 cards of the same suit.
Let C1 and C2 be independent events with P(C1) = 0.6 and P(C2) = 0.3. Compute (a) P(C1 ∩ C2) (b) P(C1 ∪ C2) (c) P(C1 ∪ Cc2).
For every two-dimensional set C contained in R2 for which the integral exists, let Q(C) = ∫ ∫C(x2 + y2) dxdy. If C1 = {(x, y) : −1 ≤ x ≤ 1,−1 ≤ y ≤ 1}, C2 = {(x, y) : −1 ≤ x = y
Let X be a random variable. If m is a positive integer, the expectation E[(X − b)m], if it exists, is called the mth moment of the distribution about the point b. Let the first, second, and third
Three distinct integers are chosen at random from the first 20 positive integers. Compute the probability that: (a) Their sum is even(b) Their product is even.
Let C denote the set of points that are interior to, or on the boundary of, a square with opposite vertices at the points (0, 0) and (1, 1). Let Q(C) = ∫ ∫C dy dx.(a) If C ⊂ C is the set {(x,
Let X be a random variable such that R(t) = E(et(X−b)) exists for t such that −h < t < h. If m is a positive integer, show that R(m)(0) is equal to the mth moment of the distribution about
There are five red chips and three blue chips in a bowl. The red chips are numbered 1, 2, 3, 4, 5, respectively, and the blue chips are numbered 1, 2, 3, respectively. If two chips are to be drawn at
Each of four persons fires one shot at a target. Let Ck denote the event that the target is hit by person k, k = 1, 2, 3, 4. If C1, C2, C3, C4 are independent and if P(C1) = P(C2) = 0.7, P(C3) = 0.9,
Let C be the set of points interior to or on the boundary of a cube with edge of length 1. Moreover, say that the cube is in the first octant with one vertex at the point (0, 0, 0) and an opposite
Let X be a random variable with mean μ and variance σ2 such that the third moment E[(X − μ)3] about the vertical line through μ exists. The value of the ratio E[(X − μ)3]/σ3 is often used
Let C denote the set {(x, y, z) : x2 + y2 + z2 ≤ 1}. Using spherical coordinates, evaluate Q(C)= +zphpap z²+z²+z²^²√ √ √ = (
Let X have the pdf f(x) = 2x, 0 < x < 1, zero elsewhere. Compute the probability that X is at least 3/4 given that X is at least 1/2.
In a lot of 50 light bulbs, there are 2 bad bulbs. An inspector examines five bulbs, which are selected at random and without replacement.(a) Find the probability of at least one defective bulb among
A bowl contains three red (R) balls and seven white (W) balls of exactly the same size and shape. Select balls successively at random and with replacement so that the events of white on the first
Let X be a random variable with mean μ and variance σ2 such that the fourth moment E[(X − μ)4] exists. The value of the ratio E[(X − μ)4]/σ4 is often used as a measure of kurtosis. Graph
A coin is tossed two independent times, each resulting in a tail (T) or a head (H). The sample space consists of four ordered pairs: TT, TH, HT, HH. Making certain assumptions, compute the
To join a certain club, a person must be either a statistician or a mathematician or both. Of the 25 members in this club, 19 are statisticians and 16 are mathematicians. How many persons in the club
A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct
After a hard-fought football game, it was reported that, of the 11 starting players, 8 hurt a hip, 6 hurt an arm, 5 hurt a knee, 3 hurt both a hip and an arm, 2 hurt both a hip and a knee, 1 hurt
A die is cast independently until the first 6 appears. If the casting stops on an odd number of times, Bob wins; otherwise, Joe wins.(a) Assuming the die is fair, what is the probability that Bob
Let X be the number of gallons of ice cream that is requested at a certain store on a hot summer day. Assume that f(x) = 12x(1000−x)2/1012, 0 < x < 1000, zero elsewhere, is the pdf of X. How
Cards are drawn at random and with replacement from an ordinary deck of 52 cards until a spade appears.(a) What is the probability that at least four draws are necessary?(b) Same as part (a), except
Suppose the experiment is to choose a real number at random in the interval (0, 1). For any subinterval (a, b) ⊂ (0, 1), it seems reasonable to assign the probability P[(a, b)] = b−a; i.e., the
A person answers each of two multiple choice questions at random. If there are four possible choices on each question, what is the conditional probability that both answers are correct given that at
Suppose a fair 6-sided die is rolled six independent times. A match occurs if side i is observed on the ith trial, i = 1, . . . , 6.(a) What is the probability of at least one match on the six
Suppose D is a nonempty collection of subsets of C. Consider the collection of events B=n{E: DCE and & is a o-field}.
Consider the events C1, C2, C3.(a) Suppose C1, C2, C3 are mutually exclusive events. If P(Ci) = pi, i = 1, 2, 3, what is the restriction on the sum p1 + p2 + p3?(b) In the notation of part (a), if p1
Let C = R, where R is the set of all real numbers. Let I be the set of all open intervals in R. The Borel σ-field on the real line is given byBy definition, B0 contains the open intervals. Because
If the pdf of X is f(x) = 2xe−x2/9, 0 < x < ∞, zero elsewhere, determine the pdf of Y = X2.
Let X have the cdf F(x) that is a mixture of the continuous and discrete types, namelyDetermine reasonable definitions of μ = E(X) and σ2 = var(X) and compute each. F(x)= 0 2+1 x < 0 #+¹ 0
Players A and B play a sequence of independent games. Player A throws a die first and wins on a “six.” If he fails, B throws and wins on a “five” or “six.” If he fails, A throws and wins
Let C1, C2, C3 be independent events with probabilities 1/2 , 1/3 , 1/4 , respectively. Compute P(C1 ∪ C2 ∪ C3).
From a bowl containing five red, three white, and seven blue chips, select four at random and without replacement. Compute the conditional probability of one red, zero white, and three blue chips,
Consider k continuous-type distributions with the following characteristics: pdf fi(x), mean μi, and variance σ2i , i = 1, 2, . . . , k. If ci ≥ 0, i = 1, 2, . . . , k, and c1+c2+· · ·+ck = 1,
Let the three mutually independent events C1, C2, and C3 be such that P(C1) = P(C2) = P(C3) = 1/4. Find P[(Cc1 ∩ Cc2) ∪ C3].
Person A tosses a coin and then person B rolls a die. This is repeated independently until a head or one of the numbers 1, 2, 3,4 appears, at which time the game is stopped. Person A wins with the
Hunters A and B shoot at a target; the probabilities of hitting the target are p1 and p2, respectively. Assuming independence, can p1 and p2 be selected so that P(zero hits) = P(one hit) = P(two
Each bag in a large box contains 25 tulip bulbs. It is known that 60% of the bags contain bulbs for 5 red and 20 yellow tulips, while the remaining 40% of the bags contain bulbs for 15 red and 10
A bowl contains 10 chips numbered 1, 2, . . . , 10, respectively. Five chips are drawn at random, one at a time, and without replacement. What is the probability that two even-numbered chips are
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