Let X1, X2, X3, and X4 be four independently selected random numbers from (0, 1). Find P (1/4 < X(3) < 1/2).
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1
Axioms of Probability
1.2
Sample Space and Events
1.4
Basic Theorems
2
Combinatorial Methods
2.2
Counting Principle
2.3
Permutations
2.4
Combinations
3
Conditional Probability and Independence
3.1
Conditional Probability
3.2
Law of Multiplication
3.3
Law of Total Probability
3.5
Independence
3.6
Applications of Probability to Genetics
4
Distribution Functions and Discrete Random Variables
4.2
Distribution Functions and Discrete Random Variables
4.4
Expectations of Discrete Random Variables
4.5
Variances and Moments of Discrete Random Variables
5
Special Discrete Distributions
5.1
Bernoulli and Binomial Random Variables
5.2
Poisson Random Variables
5.3
Other Discrete Random Variables
6
Continuous Random Variables
6.1
Probability Density Functions
6.2
Density Function of a Function of a Random Variable
6.3
Expectations and Variances
7
Special Continuous Distributions
7.2
Normal Random Variables
7.3
Exponential Random Variables
7.5
Beta Distributions
7.6
Survival Analysis and Hazard Functions
8
Bivariate Distributions
8.1
Joint Distributions of Two Random Variables
8.2
Independent Random Variables
8.3
Conditional Distributions
8.4
Transformations of Two Random Variables
9
Multivariate Distributions
9.1
Joint Distributions of n > 2 Random Variables
9.2
Order Statistics
9.3
Order Statistics
10
More Expectations and Variances
10.1
Expected Values of Sums of Random Variables
10.2
Covariance
10.3
Correlation
10.4
Conditioning on Random Variables
10.5
Bivariate Normal Distribution
11
Sums of Independent Random Variables and Limit Theorems
11.1
Moment-Generating Functions
11.2
Sums of Independent Random Variables
11.3
Markov and Chebyshev Inequalities
11.4
Laws of Large Numbers
12
Stochastic Processes
12.2
More on Poisson Processes
12.3
Markov Chains
12.4
Continuous-Time Markov Chains
12.5
Brownian Motion
13
Simulation
13.2
Simulation of Combinatorial Problems
13.4
Simulation of Random Variables
13.5
Monte Carlo Method
Textbook Solutions for Fundamentals of Probability, with Stochastic Processes
Chapter 9.2 Problem 4
Question
Let X1, X2, X3, ... , Xn be a sequence of nonnegative, identically distributed, and independent random variables. Let F be the probability distribution function of Xi, 1 i n. Prove that E[X(n)] = E 0 ! 1 4 F (x)5n" dx. Hint: Use Theorem 6.2.
Solution
The first step in solving 9.2 problem number 4 trying to solve the problem we have to refer to the textbook question: Let X1, X2, X3, ... , Xn be a sequence of nonnegative, identically distributed, and independent random variables. Let F be the probability distribution function of Xi, 1 i n. Prove that E[X(n)] = E 0 ! 1 4 F (x)5n" dx. Hint: Use Theorem 6.2.
From the textbook chapter Order Statistics you will find a few key concepts needed to solve this.
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Title
Fundamentals of Probability, with Stochastic Processes 3
Author
Saeed Ghahramani
ISBN
9780131453401
Let X1, X2, X3, ... , Xn be a sequence of nonnegative,
Chapter 9.2 textbook questions
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Chapter 9: Problem 1 Fundamentals of Probability, with Stochastic Processes 3
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Chapter 9: Problem 2 Fundamentals of Probability, with Stochastic Processes 3
Two random points are selected from (0, 1) independently. Find the probability that one of them is at least three times the other.
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Chapter 9: Problem 3 Fundamentals of Probability, with Stochastic Processes 3
Let X1, X2, X3, and X4 be independent exponential random variables, each with parameter . Find P (X(4) 3).
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Chapter 9: Problem 4 Fundamentals of Probability, with Stochastic Processes 3
Let X1, X2, X3, ... , Xn be a sequence of nonnegative, identically distributed, and independent random variables. Let F be the probability distribution function of Xi, 1 i n. Prove that E[X(n)] = E 0 ! 1 4 F (x)5n" dx. Hint: Use Theorem 6.2.
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Chapter 9: Problem 5 Fundamentals of Probability, with Stochastic Processes 3
Let X1, X2, X3, ... , Xm be a sequence of nonnegative, independent binomial random variables, each with parameters(n, p). Find the probability mass function of X(i), 1 i m.
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Chapter 9: Problem 6 Fundamentals of Probability, with Stochastic Processes 3
Prove that G, the probability distribution function of [X(1)+X(n)] = 2, the midrange of a random sample of size n from a population with continuous probability distribution function F and probability density function f , is given by G(t) = n E t 4 F (2t x) F (x)5n1 f (x) dx. Hint: Use Theorem 9.6 to find f1n; then integrate over the region x + y 2t and x y.
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Chapter 9: Problem 7 Fundamentals of Probability, with Stochastic Processes 3
Let X1 and X2 be two independent exponential random variables each with parameter . Show that X(1) and X(2) X(1) are independent.
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Chapter 9: Problem 8 Fundamentals of Probability, with Stochastic Processes 3
Let X1 and X2 be two independent N (0, 2) random variables. Find E[X(1)]. Hint: Let f12(x, y) be the joint probability density function of X(1) and X(2). The desired quantity is ## xf12(x, y) dx dy, where the integration is taken over an appropriate region.
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Chapter 9: Problem 9 Fundamentals of Probability, with Stochastic Processes 3
Let X1, X2, . . . , Xn be a random sample of size n from a population with continuous probability distribution function F and probability density function f . (a) Calculate the probability density function of the sample range, R = X(n) X(1). (b) Use (a) to find the probability density function of the sample range of n random numbers from (0, 1).
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Chapter 9: Problem 10 Fundamentals of Probability, with Stochastic Processes 3
Let X1, X2, . . . , Xn be n independently randomly selected points from the interval (0, ), > 0. Prove that E(R) = n 1 n + 1 , where R = X(n) X(1) is the range of these points. Hint: Use part (a) of Exercise 9. Also compare this with Exercise 18, Section 9.1.
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