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statistical techniques in business
Questions and Answers of
Statistical Techniques in Business
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)].
Let X1, X2 be iid with common distribution having the pdf f(x) = e−x, 0 < x < ∞, zero elsewhere. Show that Z = X1/X2 has an F-distribution.
Using the computer, obtain plots of beta pdfs for α = 1, 5, 10 and β = 1, 2, 5, 10, 20.
Let X have a binomial distribution with parameters n and p = 1/3. Determine the smallest integer n can be such that P(X ≥ 1) ≥ 0.85.
If X is N(1, 4), compute the probability P(1< X2 < 9).
Let X have a Poisson distribution with parameter m. If m is an experimental value of a random variable having a gamma distribution with α = 2 and β = 1, compute P(X = 0, 1, 2).
Suppose X is distributed N2(μ,Σ). Determine the distribution of the random vector (X1+X2,X1−X2). Show that X1+X2 and X1−X2 are independent if Var(X1) = Var(X2).
Let X1 and X2 have a trinomial distribution. Differentiate the moment generating function to show that their covariance is −np1p2.
If a fair coin is tossed at random five independent times, find the conditional probability of five heads given that there are at least four heads.
Show, for k = 1, 2, . . ., n, thatThis demonstrates the relationship between the cdfs of the β and binomial distributions. S P k-1 1²²-¹ (1-2)"-k dz = [ (") p² (1 − p)ª-². T=0 n! (k-1)!(n-k)!
Determine the constant c in each of the following so that each f(x) is a β pdf:(a) f(x) = cx(1 − x)3, 0 < x < 1, zero elsewhere.(b) f(x) = cx4(1 − x)5, 0 < x < 1, zero elsewhere.(c)
Let an unbiased die be cast at random seven independent times. Compute the conditional probability that each side appears at least once given that side 1 appears exactly twice.
Let the random variable X have a distribution that is N(μ, σ2).(a) Does the random variable Y = X2 also have a normal distribution?(b) Would the random variable Y = aX + b, a and b nonzero
Determine the constant c so that f(x) = cx(3 − x)4, 0 < x < 3, zero elsewhere, is a pdf.
Let X equal the number of independent tosses of a fair coin that are required to observe heads on consecutive tosses. Let un equal the nth Fibonacci number, where u1 = u2 = 1 and un = un−1 +
Let the random variable X be N(μ, σ2). What would this distribution be if σ2 = 0?
Let Y have a truncated distribution with pdf g(y) = φ(y)/[Φ(b)−Φ(a)], for a < y < b, zero elsewhere, where φ(x) and Φ(x) are, respectively, the pdf and distribution function of a
Let X1 and X2 be independent random variables. Let X1 and Y = X1+X2 have chi-square distributions with r1 and r degrees of freedom, respectively. Here r1 < r. Show that X2 has a chi-square
Suppose X is a random variable with the pdf f(x) which is symmetric about 0; i.e., f(−x) = f(x). Show that F(−x) = 1 − F(x), for all x in the support of X.
Consider a standard deck of 52 cards. Let X equal the number of aces in a sample of size 2.(a) If the sampling is with replacement, obtain the pmf of X.(b) If the sampling is without replacement,
Assuming a computer is available, plot the pdfs of the random variables defined in parts (a)–(d) of the last exercise. Obtain an overlay plot of all four pdfs also. In R the domain values of the
Consider a shipment of 1000 items into a factory. Suppose the factory can tolerate about 5% defective items. Let X be the number of defective items in a sample without replacement of size n = 10.
A certain job is completed in three steps in series. The means and standard deviations for the steps are (in minutes)Assuming independent steps and normal distributions, compute the probability that
Compute P(X1 + 2X2 − 2X3 > 7) if X1,X2,X3 are iid with common distribution N(1, 4).
Twenty motors were put on test under a high-temperature setting. The lifetimes in hours of the motors under these conditions are given below. Suppose we assume that the lifetime of a motor under
Let X be N(0,1). Use the moment generating function technique to show that Y = X2 is χ2(1).
Suppose the number of customers X that enter a store between the hours 9:00 a.m. and 10:00 a.m. follows a Poisson distribution with parameter θ. Suppose a random sample of the number of customers
Let X, Y , and Z have the joint pdfwhere −∞ < x < ∞, −∞ < y < ∞, and −∞ < z < ∞. While X, Y , and Z are obviously dependent, show that X, Y , and Z are pairwise
The monthly linear trend equation for the Hoopes ABC Beverage Store is:ŷ = 5.50 + 1.25tThe equation is based on 4 years of monthly data and is reported in thousands of dollars. The index for January
Listed below are the price and quantity of several golf items purchased by members of the men’s golf league at Indigo Creek Golf and Tennis Club for 2012 and 2016.a. Determine the simple aggregate
Listed below are the sales at Roberta’s Ice Cream Stand for the last 5 years, 2012 through 2016.Year....................................................
Based on five years of monthly data (the period from January 2011 to December 2015), the trend equation for a small company is ŷ = 3.5 + 0.7t. The seasonal index for January is 120 and for June it
According to a study in Health Magazine, one in three children in the United States is obese or overweight. A health practitioner in Louisiana sampled 500 children and found 210 who were obese or
An instructor has three sections of basic statistics: 8:00 a.m., 10:00 a.m., and 1:30 p.m. Listed ?below are the grades on the first exam for each section. Assume that the distributions do not follow
A book publisher wants to investigate the type of book selected for recreational reading by men and women. A random sample of 540 men and 500 women provided the following information regarding their
A recent census report indicated that 65% of families have two parents present, 20% have only ?mother present, 10% have only a father present, and 5% have no parent present. A random sample of 200
Given the following regression analysis output.a. What is the sample size?b. How many independent variables are in the study?c. Determine the coefficient of determination.d. Conduct a global test of
Given the following regression analysis output:a. What is the sample size?b. Write out the regression equation. Interpret the slope and intercept values.c. If the value of the independent variable is
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