Questions and Answers of Statistical Techniques in Business

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
Let X1,X2,X3 be iid, each with the distribution having pdf f(x) = e−x, 0 < x < ∞, zero elsewhere. Show thatare mutually independent. Y₁ X₁ X₁ + X₂¹ Y₂ = X₁ + X₂ X₁ + X2 +
A person bets 1 dollar to b dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. Find b so that the bet is fair.
Suppose there are three curtains. Behind one curtain there is a nice prize, while behind the other two there are worthless prizes. A contestant selects one curtain at random, and then Monte Hall
Show that the random variables X1 and X2 with joint pdfare independent. › = { ő 12x12 (12) 0
A French nobleman, Chevalier de M´er´e, had asked a famous mathematician, Pascal, to explain why the following two probabilities were different (the difference had been noted from playing the game
A chemist wishes to detect an impurity in a certain compound that she is making. There is a test that detects an impurity with probability 0.90; however, this test indicates that an impurity is there
Let the random variables X and Y have the joint pmf(a) p(x, y) = 1/3, (x, y) = (0, 0), (1, 1), (2, 2), zero elsewhere.(b) p(x, y) = 1/3, (x, y) = (0, 2), (1, 1), (2, 0), zero elsewhere.(c) p(x, y) =
Let f(x1, x2, x3) = exp[−(x1 + x2 + x3)], 0 < x1 < ∞, 0 < x2 < ∞, 0 < x3 < ∞, zero elsewhere, be the joint pdf of X1, X2, X3.(a) Compute P(X1 < X2 < X3) and P(X1 = X2
Let f1|2(x1|x2) = c1x1/x22 , 0 < x1 < x2, 0 < x2 < 1, zero elsewhere, and f2(x2) = c2x42 , 0 < x2 < 1, zero elsewhere, denote, respectively, the conditional pdf of X1, given X2 =
Let F(x, y) be the distribution function of X and Y . For all real constants a < b, c < d, show that P(a < X ≤ b, c < Y ≤ d) = F(b, d) − F(b, c) − F(a, d) + F(a, c).
Let X1 and X2 be two independent random variables so that the variances of X1 and X2 are σ21 = k and σ22 = 2, respectively. Given that the variance of Y = 3X2 − X1 is 25, find k.
Let f(x, y) = 2, 0 < x < y, 0 < y < 1, zero elsewhere, be the joint pdf of X and Y . Show that the conditional means are, respectively, (1+x)/2, 0 < x < 1, and y/2, 0 < y < 1.
Show that the function F(x, y) that is equal to 1 provided that x + 2y ≥ 1, and that is equal to zero provided that x+2y < 1, cannot be a distribution function of two random variables.
Let the joint pdf of X and Y be given by(a) Compute the marginal pdf of X and the conditional pdf of Y , given X = x.(b) For a fixed X = x, compute E(1 + x + Y |x) and use the result to compute E(Y
If the independent variables X1 and X2 have means μ1, μ2 and variances σ21, σ22, respectively, show that the mean and variance of the product Y = X1X2 are μ1μ2 and σ21σ22 + μ21σ22 +
Suppose X1 and X2 are random variables of the discrete type which have the joint pmf p(x1, x2) = (x1 + 2x2)/18, (x1, x2) = (1,1), (1,2), (2,1), (2,2), zero elsewhere. Determine the conditional mean
If f(x1, x2) = e−x1−x2 , 0 < x1 < ∞, 0 < x2 < ∞, zero elsewhere, is the joint pdf of the random variables X1 and X2, show that X1 and X2 are independent and that M(t1, t2) = (1
Let f(x, y) = e−x−y, 0 < x < ∞, 0 < y < ∞, zero elsewhere, be the pdf of X and Y. Then if Z = X +Y , compute P(Z ≤ 0), P(Z ≤ 6), and, more generally, P(Z ≤ z), for 0 < z
Let X1,X2, and X3 be three random variables with means, variances, and correlation coefficients, denoted by μ1, μ2, μ3; σ21, σ22, σ23; and ρ12, ρ13, ρ23, respectively. For constants b2 and
Determine the mean and variance of the sample mean ¯X = 5−1Σ5i=1 Xi, where X1, . . . , X5 is a random sample from a distribution having pdf f(x) = 4x3, 0 < x < 1, zero elsewhere.
Let X and Y have the joint pdf f(x, y) = 1, −x < y < x, 0 < x < 1, zero elsewhere. Show that, on the set of positive probability density, the graph of E(Y |x) is a straight line,
Let 13 cards be taken, at random and without replacement, from an ordinary deck of playing cards. If X is the number of spades in these 13 cards, find the pmf of X. If, in addition, Y is the number
Let X and Y be independent random variables with means μ1, μ2 and variances σ21, σ22. Determine the correlation coefficient of X and Z = X − Y in terms of μ1, μ2, σ21, σ22.
Let X and Y have the joint pdf f(x, y) = 3x, 0 < y < x < 1, zero elsewhere. Are X and Y independent? If not, find E(X|y).
Let X1,X2,X3 be iid with common pdf f(x) = exp(−x), 0 < x < ∞, zero elsewhere. Evaluate:(a) P(X1 < X2|X1 < 2X2).(b) P(X1 < X2 < X3|X3 < 1).
Let μ and σ2 denote the mean and variance of the random variable X. Let Y = c+ bX, where b and c are real constants. Show that the mean and variance of Y are, respectively, c + bμ and b2σ2.
Let σ21 = σ22 = σ2 be the common variance of X1 and X2 and let ρ be the correlation coefficient of X1 and X2. Show for k > 0 that P(X1-1) + (X2-μ₂)| ≥ko] ≤ 2(1 + p) k2
Determine the correlation coefficient of the random variables X and Y if var(X) = 4, var(Y) = 2, and var(X + 2Y) = 15.
Let X and Y be random variables with the space consisting of the four points (0, 0), (1, 1), (1, 0), (1,−1). Assign positive probabilities to these four points so that the correlation coefficient
Let X and Y be random variables with means μ1, μ2; variances σ21, σ22; and correlation coefficient ρ. Show that the correlation coefficient of W = aX+b, a > 0, and Z = cY + d, c > 0, is ρ.
Two line segments, each of length two units, are placed along the x-axis. The midpoint of the first is between x = 0 and x = 14 and that of the second is between x = 6 and x = 20. Assuming
A person rolls a die, tosses a coin, and draws a card from an ordinary deck. He receives $3 for each point up on the die, $10 for a head and $0 for a tail, and $1 for each spot on the card (jack =
Cast a fair die and let X = 0 if 1, 2, or 3 spots appear, let X = 1 if 4 or 5 spots appear, and let X = 2 if 6 spots appear. Do this two independent times, obtaining X1 and X2. Calculate P(|X1 −
Let X1,X2 be two random variables with joint pdf f(x1, x2) = x1 exp{−x2}, for 0 < x1 < x2 < ∞, zero elsewhere. Determine the joint mgf of X1,X2. Does M(t1, t2) = M(t1, 0)M(0, t2)?
If M(t1, t2) is the mgf of a bivariate normal distribution, compute the covariance by using the formulaNow let ψ(t1, t2) = log M(t1, t2). Show that ∂2ψ(0, 0)/∂t1∂t2 gives this covariance
The mgf of a random variable X is ( 2/3 + 1/3 et)9. Show that r=1 ( 1 ) ¸ (²) (3) 3 = ( 08 + ¹1 > x > °7 - id I 9-x
If X is b(n, p), show that E (A) = P and E[(4->)]; Ε P p(1-P) n
Let X and Y have the parameters μ1, μ2, σ21, σ22, and ρ. Show that the correlation coefficient of X and [Y − ρ(σ2/σ1)X] is zero.
Let S2 be the sample variance of a random sample from a distribution with variance σ2 > 0. Since E(S2) = σ2, why isn’t E(S) = σ?
If X is χ2(5), determine the constants c and d so that P(c < X < d) = 0.95 and P(X < c) = 0.025.
The mgf of a random variable X is e4(et−1). Show that P(μ − 2σ < X < μ + 2σ) = 0.931.
Let X have the conditional geometric pmf θ(1−θ)x−1, x = 1, 2, . . ., where θ is a value of a random variable having a beta pdf with parameters α and β. Show that the marginal (unconditional)
In a lengthy manuscript, it is discovered that only 13.5 percent of the pages contain no typing errors. If we assume that the number of errors per page is a random variable with a Poisson
Let U and V be independent random variables, each having a standard normal distribution. Show that the mgf E(et(UV)) of the random variable UV is (1 − t2)−1/2, −1 < t < 1.
Assuming a computer is available, plot the pdfs of the random variables defined in parts (a)–(e) below. Obtain an overlay plot of all five pdfs, also. In R the domain values of the pdfs can easily
Suppose that g(x, 0) = 0 and thatfor x = 1, 2, 3, . . .. If g(0, w) = e−λw, show by mathematical induction that Du g(x, w)]=-Ag(x, w) + Ag(x-1, w)
Let X be a random variable such that E(Xm) = (m+1)!2m, m = 1, 2, 3, . . . . Determine the mgf and the distribution of X.
Let X have a Poisson distribution with μ = 100. Use Chebyshev’s inequality to determine a lower bound for P(75 < X < 125).
Let X and Y have a bivariate normal distribution with parameters μ1 = 5, μ2 = 10, σ21 = 1, σ22 = 25, and ρ > 0. If P(4 < Y < 16|X = 5) = 0.954, determine ρ.
Assuming a computer is available, investigate the probabilities of an “outlier” for a t-random variable and a normal random variable. Specifically, determine the probability of observing the
Let Y be the number of successes in n independent repetitions of a random experiment having the probability of success p = 2/3. If n = 3, compute P(2 ≤ Y); if n = 5, compute P(3 ≤ Y).
Let Y be the number of successes throughout n independent repetitions of a random experiment with probability of success p = 1/4. Determine the smallest value of n so that P(1 ≤ Y ) ≥ 0.70.
Using the computer, obtain an overlay plot of the pmfs following two distributions:(a) Poisson distribution with λ = 2.(b) Binomial distribution with n = 100 and p = 0.02.Why would these
Let the independent random variables X1 and X2 have binomial distribution with parameters n1 = 3, p = 2/3 and n2 = 4, p = 1/2 , respectively. Compute P(X1 = X2).
Compute the measures of skewness and kurtosis of a gamma distribution which has parameters α and β.
Compute the measures of skewness and kurtosis of the Poisson distribution with mean μ.
Let X have a gamma distribution with parameters α and β. Show that P(X ≥ 2αβ) ≤ (2/e)α.
Toss two nickels and three dimes at random. Make appropriate assumptions and compute the probability that there are more heads showing on the nickels than on the dimes.
Determine the 90th percentile of the distribution, which is N(65, 25).
On the average, a grocer sells three of a certain article per week. How many of these should he have in stock so that the chance of his running out within a week is less than 0.01? Assume a Poisson
Let T = W/√V/r, where the independent variables W and V are, respectively, normal with mean zero and variance 1 and chi-square with r degrees of freedom. Show that T2 has an F-distribution with
Using the computer, obtain plots of the pdfs of chi-squared distributions with degrees of freedom r = 1, 2, 5, 10, 20. Comment on the plots.
Using the computer, plot the cdf of Γ(5, 4) and use it to guess the median. Confirm it with a computer command which returns the median [In R, use the command q gamma(.5,shape=5,scale=4)].

Showing 700 - 800 of 1142