Questions and Answers of Introduction To Mathematical Statistics

Evaluate the integral using the Student t distribution. Jo ∞ 1 1+x² dx
Let k denote the number of successes observed in a sequence of n independent Bernoulli trials, where p = P(success). (a) Show that the critical region of the likelihood ratio test of H0: p = 1/2
Evaluate the following probabilities: (a) (b) (c) (d) ,2 P(x17 ≥ 8.672)
Let y1, y2,..., yn be a random sample from a normal pdf with unknown mean μ and variance 1. Find the form of the GLRT for H0:μ = μ0 versus H1:μ ≠ μo.
A sample of size 1 from the pdf fY (y) = (1 + θ)yθ, 0 ≤ y ≤ 1 and θ > −1, is to be the basis for testingThe critical region will be the interval y ≤ 1/2 . Find an expression for 1 −
Let Y1,Y2,...,Yn be a random sample of size n from a normal distribution having mean μ and variance σ2. What is the smallest value of n for which the following is true? P 0² 0.95
In a nongeriatric population, platelet counts ranging from 140 to 440 (thousands per mm3 of blood) are considered “normal.” The following are the platelet counts recorded for twenty-four female
If H0:μ = μ0 is rejected in favor of H1:μ > μ0, will it necessarily be rejected in favor of H1:μ ≠ μ0? Assume that α remains the same.
For a Student t random variable T with n degrees of freedom and any positive integer k, show that E(T2k) exists if 2k < n.are finite if α > 0, β > 0, and αβ > 1.) 5.° 1 (1+ya)³ dy
Historically, fluctuations in the amount of rare metals found in coins are not uncommon (82). The following data may be a case in point. Listed are the silver percentages found in samples of a
Would the following residual plot produced by fitting a least squares straight line to a set of n = 13 points cause you to doubt that the underlying xy-relationship is linear? Explain.
An Atomic Energy Commission nuclear facility was established in Hanford, Washington, in 1943. Over the years, a significant amount of strontium 90 and cesium 137 leaked into the Columbia River. In a
What, if anything, is unusual about the following residual plots? Residual, y-ý Residual, y-ý
Let (x1, y1), (x2, y2), ..., (xn, yn) be a set of measurements whose sample correlation coefficient is r. Show thatwhere βˆ1 is the maximum likelihood estimate for the slope. r = B₁ n ηΣ
The growth of federal expenditures is one of the characteristic features of the U.S. economy. The rapidity of the increases from 2000 to 2015, as shown in the table below, suggests an exponential
Listed below are forty ordered computergenerated observations that presumably represent a normal distribution with μ = 5 and σ = 2. Can the sample be considered random with respect to the number
Prove that the variance of Yˆ can also be written Var(f) – n σ'Σ (x − x) i=1 n nΣ (x − x)2 i=1
(a) For random variables X and Y, show that Cov(X + Y, X − Y) = Var(X) − Var(Y) (b) Suppose that Cov(X,Y) = 0. Prove that ρ(X + Y, X − Y) = Var(X) − Var(Y)/Var(X) + Var(Y)
Suppose that Xi is a random variable for which E(Xi) = μ ≠ 0, i = 1, 2,..., n. Under what conditions will the following be true? n E Ε(ΣαX; Χ \i=1 μ
How many terms will be included in the expansion of (a + b + c) (d + e + f) (x + y + u + v + w) Which of the following will be included in that number: aeu, cdx, bef, xvw?
Let Y be a continuous random variable with fY (y) = 1/2 (1+y), −1 ≤ y ≤ 1. Define the random variable W by W = −4Y +7. Find fW (w). Be sure to specify those values of w for which fW (w) ≠ 0.
Let fY (y) = 3/14 (1 + y2), 0 ≤ y ≤ 2. Define the random variable W by W = 3Y + 2. Find fW (w). Be sure to specify the values of w for which fW (w) = 0.
Let Y have pdf Find MY (t). [y, fy(y)=2-y, 0, 0 ≤ y ≤ 1 1≤ y ≤2 elsewhere
Suppose that fX,Y (x, y) = λ2e−λ(x+y) , 0 ≤ x, 0 ≤ y. Find E(X + Y).
Suppose that random variables X and Y vary in accordance with the joint pdf, fX,Y (x, y) = c(x+y), 0 < x < y < 1. Find c.
If the pdf for Y is find and graph FY (y). f» = { = 0, ]y[ > 1 [ 1 - ]y[, _ ]y[
Suppose fX (x) = xe−x, x ≥ 0, and fY (y) = e−y, y ≥ 0, where X and Y are independent. Find the pdf of X + Y.
Let Y be a random variable with fY (y) = 6y(1−y), 0 ≤ y ≤ 1. Find the pdf of W = Y2.
Let X be the time in days between a car accident and reporting a claim to the insurance company. Let Y be the time in days between the report and payment of the claim. Suppose that fX,Y (x, y) = c, 0
Let X and Y have the joint pdf fX,Y (x, y) = 2e−(x+y) , 0 < x < y, 0 < yFind P(Y < 3X).
A random variable Y has cdf Find(a) P(Y < 2)(b) P (2 < Y ≤ 2 1/2) (c) P ( 2 < Y < 2 1/2)(d) fY (y) 0 Fy (y) = In y In y 1 y < 1 1 ≤ y ≤e e < y
Suppose that fX,Y (x, y) = λ2e−λ(x+y) , 0 ≤ x, 0 ≤ y. Find Var(X + Y).
Suppose that fX,Y (x, y) = 3/2 (x2 + y2), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Find Var(X + Y).
For continuous random variables X and Y, prove that E(Y) = EX [E(Y|x)].
Consider an urn with r red balls and w white balls, where r + w = N. Draw n balls in order without replacement. Show that the probability of k red balls is hypergeometric.
A typical day’s production of a certain electronic component is twelve. The probability that one of these components needs rework is 0.11. Each component needing rework costs $100. What is the
Let Y have probability density function fY (y) = 2(1 − y), 0 ≤ y ≤ 1Suppose that W = Y2, in which caseFind E(W) in two different ways. fw (w) = - 1 W 1, 0≤ w ≤ 1
Calculate PX,Y (0, 1) if px,y,z(x, y, z) = 3-x-y-z ( ) ( ) ( )² · () ³-*-y-² for x, y, z = 0, 1, . 3. 3! x!y!z!(3-x-y-z)! 2, 3 and 0≤x+y+z≤
A random sample of size 8—x1 = 1, x2 = 0, x3 = 1, x4 = 1, x5 = 0, x6 = 1, x7 = 1, and x8 = 0—is taken from the probability functionFind the maximum likelihood estimate for θ. Px (k; 0) = 0k (1 -
Two fair dice are tossed. Let X denote the number appearing on the first die and Y the number on the second. Show that X and Y are independent.
Suppose that random variables X and Y are independent with marginal pdfs fX (x) = 2x, 0 ≤ x ≤ 1, and fY (y) = 3y2, 0 ≤ y ≤ 1. Find P(Y < X).
Write down the joint probability density function for a random sample of size n drawn from the exponential pdf, fX (x) = (1/λ)e−x/λ, x ≥ 0.
Suppose Y1,Y2,...,Yn is a random sample from the exponential pdf, fY (y; λ) = λe−λy, y > 0.(a) Show that λˆn = Y1 is not consistent for λ.(b) Show that is not consistent for λ.
Suppose θˆn = Ymax is to be used as an estimator for the parameter θ in the uniform pdf, fY (y; θ) = 1/θ, 0 ≤ y ≤ θ. Show that θˆn is squared-error consistent (see Question 5.7.4).Data in
Suppose that X1, X2, X3, and X4 are independent random variables, each with pdf fXi (xi) = 4x3i , 0 ≤ xi ≤ 1. Find(a) P (X1 < 1/2).(b) P (exactly one Xi < 1/2).(c) fX1,X2,X3,X4 (x1, x2, x3,
Suppose Y1,Y2,...,Yn is a random sample from the exponential pdf, fY (y; λ) = λe−λy, y > 0.(a) Show that λˆn = Y1 is not consistent for λ.(b) Show that is not consistent for λ.
According to an airline industry report (189), roughly one piece of luggage out of every two hundred that are checked is lost. Suppose that a frequent-flying businesswoman will be checking one
Show that is sufficient for σ2 if Y1,Y2,...,Yn is a random sample from a normal pdf with μ = 0. n 62 = ΣΥ = i=1
Use the method of maximum likelihood to estimate θ in the pdf Evaluate θe for the following random sample of size 4: y1 = 6.2, y2 = 7.0, y3 = 2.5, and y4 = 4.2. fy (y; 0)
If θˆ is sufficient for θ, show that any one-to-one function of θˆ is also sufficient for θ.
Let Y1,Y2,...,Yn be a random sample of size n from the pdf(a) Show that θˆ = 1/rY is an unbiased estimator for θ.(b) Show that θˆ = 1/rY is a minimum-variance estimator for θ. fy (y; 0)
Given that y1 = 2.3, y2 = 1.9, and y3 = 4.6 is a random sample fromcalculate the maximum likelihood estimate for θ. fy (y; 0) = y³e-y/0 604 y≥0
Use the sample y1 = 8.2, y2 = 9.1, y3 = 10.6, and y4 = 4.9 to calculate the maximum likelihood estimate for λ in the exponential pdf fY (y; λ) = λe−λy , y ≥ 0
A pdf gW (w; θ) is said to be expressed in exponential form if it can be written as where the range of W is independent of θ. Show that is sufficient for θ. gw (w; 0) =
A random sample of size 2,Y1 and Y2, is drawn from the pdfWhat must c equal if the statistic c(Y1 + 2Y2) is to be an unbiased estimator for 1/θ ? 1 fy (y; 0) = 2y0², 0
Find the squared-error loss Bayes estimate for θ in Question 5.8.7. Data in Question 5.8.7.Let Y1,Y2,...,Yn be a random sample from a gamma pdf with parameters r and θ, where the prior
Let Y1,Y2,...,Yn be a random sample from a Pareto pdf, fY (y; θ) = θ/(1 + y) θ+1 , 0 ≤ y ≤ ∞; 0 < θ < ∞. Write fY (y; θ) in exponential form and deduce a sufficient statistic
A sample of size 1 is drawn from the uniform pdf defined over the interval [0, θ]. Find an unbiased estimator for θ2.
Examine the first two derivatives of the function g(p) = p(1−p) to verify the claim on p. 304 that p(1−p) ≤ 1/4 for 0 < p < 1.
Show that n(n − 1)2n–² = Σ k(k − 1)("). – - k=2
A restaurant offers a choice of four appetizers, fourteen entrees, six desserts, and five beverages. How many different meals are possible if a diner intends to order only three courses? (Consider
In international Morse code, each letter in the alphabet is symbolized by a series of dots and dashes: the letter a, for example, is encoded as “· –”. What is the minimum number of dots and/or
If the letters in the phrase are arranged at random, what are the chances that not all the S’s will be adjacent? A ROLLING STONE GATHERS NO MOSS
Consider the following four-switch circuit: If all switches operate independently and P(Switch closes) = p, what is the probability the circuit is completed? In A₁ A3 A₂ A4 Out
The board of a large corporation has six members willing to be nominated for office. How many different “president/vice president/treasurer” slates could be submitted to the stockholders?
One chip is drawn at random from an urn that contains one white chip and one black chip. If the white chip is selected, we simply return it to the urn; if the black chip is drawn, that
A new horror movie, Friday the 13th, Part X, will star Jason’s great-grandson (also named Jason) as a psychotic trying to dispatch (as gruesomely as possible) eight camp counselors, four men and
What must be true of events A and B if (a) A ∪ B = B(b) A ∩ B = A
A computer is instructed to generate a random sequence using the digits 0 through 9; repetitions are permissible. What is the shortest length the sequence can be and still have at least a 70%
Emma and Josh have just gotten engaged. What is the probability that they have different blood types? Assume that blood types for both men and women are distributed in the general population
A chemical engineer wishes to observe the effects of temperature, pressure, and catalyst concentration on the yield resulting from a certain reaction. If she intends to include two different
Find P(A ∩ B) if P(A) = 0.2, P(B) = 0.4, and P(A|B) + P(B|A) = 0.75.
Given that P(A) = a and P(B) = b, show that P(A/B) > IV a+b-1 b
Express the following probabilities in terms of P(A), P(B), and P(A ∩ B).(a) P(AC ∪ BC)(b) P(AC ∩ (A ∪ B))
If P(A|B) < P(A), show that P(B|A) < P(B).
An urn contains six chips numbered 1 through 6. Three are drawn out. What outcomes are in the event “Second smallest chip is a 3”? Assume that the order of the chips is irrelevant.
Three points, X1, X2, and X3, are chosen at random in the interval (0, a). A second set of three points, Y1, Y2, and Y3, are chosen at random in the interval (0, b). Let A be the event that X2 is
A fast-food restaurant offers customers a choice of eight toppings that can be added to a hamburger. How many different hamburgers can be ordered?
Suppose a baseball player steps to the plate with the intention of trying to “coax” a base on balls by never swinging at a pitch. The umpire, of course, will necessarily call each pitch either a
In the game of “odd man out” each player tosses a fair coin. If all the coins turn up the same except for one, the player tossing the different coin is declared the odd man out and is eliminated
Six dice are rolled one time. What is the probability that each of the six faces appears?
Given that P(A ∩ BC) = 0.3, P((A ∪ B) C) = 0.2, and P(A ∩ B) = 0.1, find P(A|B).
Suppose that two cards are drawn—in order— from a standard 52-card poker deck. In how many ways can the first card be a club and the second card be an ace?
Given that P(A) + P(B) = 0.9, P(A|B) = 0.5, and P(B|A) = 0.4, find P(A).
Monica’s vacation plans require that she fly from Nashville to Chicago to Seattle to Anchorage. According to her travel agent, there are three available flights from Nashville to Chicago, five from
Let A, B, and C be three events defined on a sample space, S. Arrange the probabilities of the following events from smallest to largest:(a) A ∪ B(b) A ∩ B(c) A(d) S(e) (A ∩ B) ∪ (A ∩ C)
How many ways can a set of four tires be put on a car if all the tires are interchangeable? How many ways are possible if two of the four are snow tires?
Find A ∩ B ∩ C if A = {x: 0 ≤ x ≤ 4}, B = {x: 2 ≤ x ≤ 6}, and C = {x: x = 0, 1, 2,...}.
The nine members of the music faculty baseball team, the Mahler Maulers, are all incompetent, and each can play any position equally poorly. In how many different ways can the Maulers take the field?
In a certain developing nation, statistics show that only two out of ten children born in the early 1980s reached the age of twenty-one. If the same mortality rate is operative over the next
Pictured below are two organizational charts describing the way upper management vets new proposals. For both models, three vice presidents—1, 2, and 3—each voice an opinion. For (a), all
Let A and B be any two events defined on a sample space S. Which of the following sets are necessarily subsets of which other sets? A B An BC AUB (ACU BC) C ANB AC N B
In how many ways can a pack of fifty-two cards be dealt to thirteen players, four to each, so that every player has one card of each suit?
A coin is tossed four times and the resulting sequence of heads and/or tails is recorded. Define the events A, B, and C as follows: A: Exactly two heads appearB: Heads and tails alternateC:
Prove that directly without appealing to any combinatorial arguments. n+1 n (" + ¹) = ( ² ) + ( x ²₁ ) k - 1
Stanley’s statistics seminar is graded on a Pass/Fail basis. At the end of the semester each student is given the option of taking either a two-question exam (Final A) or a three-question exam
The crew of Apollo 17 consisted of a pilot, a copilot, and a geologist. Suppose that NASA had actually trained nine aviators and four geologists as candidates for the flight. How many different crews
What is the smallest number of switches wired in parallel that will give a probability of at least 0.98 that a circuit will be completed? Assume that each switch operates independently and will
In an upstate congressional race, the incumbent Republican (R) is running against a field of three Democrats (D1, D2, and D3) seeking the nomination. Political pundits estimate that the probabilities
Which state name can generate more permutations, TENNESSEE or FLORIDA?

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