A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 10, Problem 10.15P
(a)
To determine
To Prove: The minimum of
(b)
To determine
To Prove: The minimum of
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The same final exam of the astronomy course was given to two groups of students. The maximum number of points that a student can score is 100. The first group consisted of a random sample of 10 students who were taught by Professor A. Students from the first group obtained the following results:
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The second group consisted of a random sample of 9 students who were taught by Professor B. Students from the second group obtained the following results:
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Compute the mean squares of between-group variability, MSBET. Round your answer to two decimal places.
1. Consider the following preference ballots:
Number of voters
Rankings 6 5 4 2
1st choice A DCB
DC
2nd choice B B D
3rd choice DCBD
4th choice CA
AAA
For each of the four voting systems we have studied, determine who would win the election
in each case. (Remember: For plurality with runoff, all but the top two vote-getters are
simultaneously eliminated at the end of round 1.)
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Chapter 10 Solutions
A First Course in Probability
Ch. 10 - The following algorithm will generate a random...Ch. 10 - Prob. 10.2PCh. 10 - Give a technique for simulating a random variable...Ch. 10 - Present a method for simulating a random variable...Ch. 10 - Use the inverse transformation method to present...Ch. 10 - Give a method for simulating a random variable...Ch. 10 - Let F be the distribution functionF(x)=xn0x1 a....Ch. 10 - Prob. 10.8PCh. 10 - Suppose we have a method for simulating random...Ch. 10 - Prob. 10.10P
Ch. 10 - Use the rejection method with g(x)=1,0x1, to...Ch. 10 - Prob. 10.12PCh. 10 - Prob. 10.13PCh. 10 - Prob. 10.14PCh. 10 - Prob. 10.15PCh. 10 - Let X be a random variable on (0, 1) whose density...Ch. 10 - Prob. 10.1STPECh. 10 - Prob. 10.2STPECh. 10 - Prob. 10.3STPECh. 10 - If X is a normal random variable with mean and...Ch. 10 - Prob. 10.5STPE
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- iid B1 Suppose X1, ..., Xn fx(x), where 2 fx(x) = x exp(−x²/0), 0<< (0 otherwise). (a) Find the maximum likelihood estimator of 0. (b) Show that the MLE is an unbiased estimator of 0. (c) Find the MSE of the MLE. Hint: For parts (b) and (c), you may use integration by parts.arrow_forwardiid B1 Suppose X1, ..., Xn fx(x), where 2 fx(x) = x exp(−x²/0), 0<< (0 otherwise). (a) Find the maximum likelihood estimator of 0. (b) Show that the MLE is an unbiased estimator of 0. (c) Find the MSE of the MLE. Hint: For parts (b) and (c), you may use integration by parts.arrow_forward2) Suppose we select two values x and y independently from the uniform distribution on [0,1]. What is the probability that xyarrow_forward100 identical balls are rolling along a straight line. They all have speed equal to v, but some of them might move in opposite directions. When two of them collide they immediately switch their direction and keep the speed v. What is the maximum number of collisions that can happen? Let f(w) be a function of vector w Є RN, i.e. f(w) = 1+e Determine the first derivative and matrix of second derivatives off with respect to w. Let A Є RN*N be a symmetric, positive definite matrix and bЄ RN a vector. If x ER, evaluate the integral Z(A,b) = e¯xAx+bx dx as a function of A and b. John throws a fair die with faces labelled 1 to 6. ⚫ He gains 10 points if the die shows 1. ⚫ He gains 1 point if the die shows 2 or 4. • No points are allocated otherwise. Let X be the random variable describing John's gain at each throw. Determine the variance of X.arrow_forwardFemale Male Totals Less than High School Diploma 0.077 0.110 0.187 High School Diploma 0.154 0.201 0.355 Some College/University 0.141 0.129 0.270 College/University Graduate 0.092 0.096 0.188 Totals 0.464 0.536 1.000arrow_forwardFemale Male Totals Less than High School Diploma 0.077 0.110 0.187 High School Diploma 0.154 0.201 0.355 Some College/University 0.141 0.129 0.270 College/University Graduate 0.092 0.096 0.188 Totals 0.464 0.536 1.000arrow_forwardFemale Male Totals Less than High School Diploma 0.077 0.110 0.187 High School Diploma 0.154 0.201 0.355 Some College/University 0.141 0.129 0.270 College/University Graduate 0.092 0.096 0.188 Totals 0.464 0.536 1.000arrow_forwardFemale Male Totals Less than High School Diploma 0.077 0.110 0.187 High School Diploma 0.154 0.201 0.355 Some College/University 0.141 0.129 0.270 College/University Graduate 0.092 0.096 0.188 Totals 0.464 0.536 1.000arrow_forward6.54 Let Y₁, Y2,..., Y, be independent Poisson random variables with means 1, 2,..., An respectively. Find the a probability function of Y. b conditional probability function of Y₁, given that Y = m. Y₁ = m. c conditional probability function of Y₁+Y2, given that 6.55 Customers arrive at a department store checkout counter according to a Poisson distribution with a mean of 7 per hour. In a given two-hour period, what is the probability that 20 or more customers will arrive at the counter? 6.56 The length of time necessary to tune up a car is exponentially distributed with a mean of .5 hour. If two cars are waiting for a tune-up and the service times are independent, what is the probability that the total time for the two tune-ups will exceed 1.5 hours? [Hint: Recall the result of Example 6.12.] 6.57 Let Y, Y2,..., Y,, be independent random variables such that each Y, has a gamma distribution with parameters a, and B. That is, the distributions of the Y's might have different a's, but…arrow_forward6.82 6.83 6.84 6.85 *6.86 6.87 If Y is a continuous random variable and m is the median of the distribution, then m is such that P(Ym) = P(Y ≥ m) = 1/2. If Y₁, Y2,..., Y, are independent, exponentially dis- tributed random variables with mean ẞ and median m, Example 6.17 implies that Y(n) = max(Y₁, Y., Y) does not have an exponential distribution. Use the general form of FY() (y) to show that P(Y(n) > m) = 1 - (.5)". Refer to Exercise 6.82. If Y₁, Y2,..., Y,, is a random sample from any continuous distribution with mean m, what is P(Y(n) > m)? Refer to Exercise 6.26. The Weibull density function is given by -my" m-le-y/a f(y)= α 0. y > 0, elsewhere, where a and m are positive constants. If a random sample of size n is taken from a Weibull distributed population, find the distribution function and density function for Y(1) = min(Y1, Y2,Y). Does Y(1) = have a Weibull distribution? Let Y₁ and Y2 be independent and uniformly distributed over the interval (0, 1). Find P(2Y(1) 0, elsewhere,…arrow_forward6.26 The Weibull density function is given by e-y/a f(y) = α 0. y > 0, elsewhere, where a and m are positive constants. This density function is often used as a model for the lengths of life of physical systems. Suppose Y has the Weibull density just given. Find a the density function of UY". b E(Y) for any positive integer k. 6.27 Let Y have an exponential distribution with mean ẞ. 6.28 6.29 a Prove that W = √Y has a Weibull density with α = ẞ and m = 2. b Use the result in Exercise 6.26(b) to give E(Yk/2) for any positive integer k. Let Y have a uniform (0, 1) distribution. Show that U = -2ln(Y) has an exponential distri- bution with mean 2. The speed of a molecule in a uniform gas at equilibrium is a random variable V whose density function is given by 6.30 6.31 6.32 f(v) = av²e-by², v > 0, where b = m/2kT and k, T, and m denote Boltzmann's constant, the absolute temperature, and the mass of the molecule, respectively. a Derive the distribution of W = mV2/2, the kinetic energy of…arrow_forwardQIA Let F-4c24, countible or, A, countible), show that is o-algebra. B Let (Fne N) is family of a-algebra on 2, prove that F. o-algebra Q2: Prove that: 1. X, is martingale -esin 2. M, -e sin B,, is martingale by using Ito formula Q3: A Let X, has stochastic differential with drift p(x)=-bx + c, and diffusion o²(x)=4x, let Y√X,, where X, ≥0, find dr B: Let X, -(-s), Ito integral process, find dx, and [x.xko). Q4: Let Y, =[x,dB, is Ito integral, such that X, is nonrandom process, find: التوزيع 1. The distribution of Y 2. The moment generating function of Y,.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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