A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 8, Problem 8.12TE
To determine
To show: The right side of Chernoff bound of the inequality is reduced by the factor 2.
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Let X1,X2 denote the variables for the two-dimensional data in Exercise 1. Find a new variable yl of the form
yı = cjX¡ + c2X2, with c +c; = 1, such that y1 has maximum possible variance over the given data.
How much of the variance in the data is explained by y1?
The built-in data set, rock, is a data frame recording the measurements of rock sample from a petroleum reservoir. We are interested in the area column of rock. Using R we can convert this column into the vector x by the assignment x<-rock$shape. Assume that the n area measurements x=(x1, x2,...,xn) are a random sample from a population with true unknown mean ? and true unknown variance ?2. Remember, let x be defined by
x<-rock$shape.
d, e , f
Chapter 8 Solutions
A First Course in Probability
Ch. 8 - Suppose that X is a random variable with mean and...Ch. 8 - From past experience, a professor knows that the...Ch. 8 - Use the central limit theorem to solve part (c) of...Ch. 8 - Let X1,...,X20 be independent Poisson random...Ch. 8 - Fifty numbers are rounded off to the nearest...Ch. 8 - A die is continually rolled until the total sum of...Ch. 8 - A person has 100 light bulbs whose lifetimes are...Ch. 8 - In Problem 8.7, suppose that it takes a random...Ch. 8 - If X is a gamma random variable with parameters...Ch. 8 - Civil engineers believe that W, the amount of...
Ch. 8 - Many people believe that the daily change of price...Ch. 8 - We have 100 components that we will put in use in...Ch. 8 - Student scores on exams given by a certain...Ch. 8 - A certain component is critical to the operation...Ch. 8 - An insurance company has 10.000 automobile...Ch. 8 - A.J. has 20 jobs that she must do in sequence,...Ch. 8 - Redo Example 5b under the assumption that the...Ch. 8 - Repeat part (a) of Problem 8.2 when it is known...Ch. 8 - A lake contains 4 distinct types of fish. Suppose...Ch. 8 - If X is a nonne9ative random variable with mean...Ch. 8 - Let X be a nonnegative random variable. Prove that...Ch. 8 - Prob. 8.22PCh. 8 - Let X be a Poisson random variable with mean 20....Ch. 8 - Prob. 8.24PCh. 8 - If X has variance 2, then , the positive square...Ch. 8 - If X has mean and standard deviation , the ratio...Ch. 8 - Compute the measurement signal-to-noise ratio-that...Ch. 8 - Let Zn,n1, be a sequence of random variables and...Ch. 8 - Prob. 8.5TECh. 8 - Prob. 8.6TECh. 8 - Prob. 8.7TECh. 8 - Explain why a gamma random variable with...Ch. 8 - Prob. 8.9TECh. 8 - If X is a Poisson random variable with mean , show...Ch. 8 - Prob. 8.11TECh. 8 - Prob. 8.12TECh. 8 - Prob. 8.13TECh. 8 - Prob. 8.14TECh. 8 - The number of automobiles sold weekly at a certain...Ch. 8 - Prob. 8.2STPECh. 8 - If E[X]=75E[Y]=75Var(X)=10var(Y)=12cov(X,Y)=3 give...Ch. 8 - Prob. 8.4STPECh. 8 - Prob. 8.5STPECh. 8 - Prob. 8.6STPECh. 8 - Prob. 8.7STPECh. 8 - Prob. 8.8STPECh. 8 - Prob. 8.9STPECh. 8 - A tobacco company claims that the amount of...Ch. 8 - Prob. 8.11STPECh. 8 - Prob. 8.12STPECh. 8 - The strong law of large numbers states that with...Ch. 8 - Each new book donated to a library must be...
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Similar questions
- Let X be a random variable and a real number. Show that E(X - a)² = varX + (µ − a)² Hereμ = EX is the expected value of the random variable X and varX = E(X - μ)^2 is the variance of the random variable X. Guidance: start from the representation - (X-a)^2 = (X µ + μ- a)^2 and group the right side of the representation appropriately into the form (Z + b)^2, where Z is some random variable and b is a real number and open the square. The task should be solved with the help of the expected value calculation rules.arrow_forwardSuppose the variance of the difference of two random variables X and Y is zero. Then, show that X = Y + c for some constant c almost surely. Also find the value of c.arrow_forwardThe built-in data set, rock, is a data frame recording the measurements of rock sample from a petroleum reservoir. We are interested in the area column of rock. Using R we can convert this column into the vector x by the assignment x30 we can create a confidence interval for μ using a normal critical value. If we want the confidence interval to be at the 94% level and we use a normal critical value, then what critical value should we use? i) Calculate a 94% confidence interval(using a normal critical value) for μ.arrow_forward
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