Prove Corollary 5.9.2.
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1.10
Introduction to Probability
1.12
Introduction to Probability
1.4
Introduction to Probability
1.5
Introduction to Probability
1.6
Introduction to Probability
1.7
Introduction to Probability
1.8
Introduction to Probability
1.9
Introduction to Probability
2.1
Conditional Probability
2.2
Conditional Probability
2.3
Conditional Probability
2.4
Conditional Probability
2.5
Conditional Probability
3.1
Random Variables and Distributions
3.10
Random Variables and Distributions
3.11
Random Variables and Distributions
3.2
Random Variables and Distributions
3.3
Random Variables and Distributions
3.4
Random Variables and Distributions
3.5
Random Variables and Distributions
3.6
Random Variables and Distributions
3.7
Random Variables and Distributions
3.8
Random Variables and Distributions
3.9
Random Variables and Distributions
4.1
Expectation
4.2
Expectation
4.3
Expectation
4.4
Expectation
4.5
Expectation
4.6
Expectation
4.7
Expectation
4.8
Expectation
4.9
Expectation
5.10
Special Distributions
5.11
Special Distributions
5.2
Special Distributions
5.3
Special Distributions
5.4
Special Distributions
5.5
Special Distributions
5.6
Special Distributions
5.7
Special Distributions
5.8
Special Distributions
5.9
Special Distributions
6.1
Large Random Samples
6.2
Large Random Samples
6.3
Large Random Samples
6.4
Large Random Samples
6.5
Large Random Samples
7.1
Estimation
7.10
Estimation
7.2
Estimation
7.3
Estimation
7.4
Estimation
7.5
Estimation
7.6
Estimation
7.7
Estimation
7.8
Estimation
7.9
Estimation
8.1
Sampling Distributions of Estimators
8.2
Sampling Distributions of Estimators
8.3
Sampling Distributions of Estimators
8.4
Sampling Distributions of Estimators
8.5
Sampling Distributions of Estimators
8.6
Sampling Distributions of Estimators
8.7
Sampling Distributions of Estimators
8.8
Sampling Distributions of Estimators
8.9
Sampling Distributions of Estimators
9.1
Testing Hypotheses
9.10
Testing Hypotheses
9.2
Testing Hypotheses
9.3
Testing Hypotheses
9.4
Testing Hypotheses
9.5
Testing Hypotheses
9.6
Testing Hypotheses
9.7
Testing Hypotheses
9.8
Testing Hypotheses
9.9
Testing Hypotheses
10.1
Categorical Data and Nonparametric Methods
10.2
Categorical Data and Nonparametric Methods
10.3
Categorical Data and Nonparametric Methods
10.4
Categorical Data and Nonparametric Methods
10.5
Categorical Data and Nonparametric Methods
10.6
Categorical Data and Nonparametric Methods
10.7
Categorical Data and Nonparametric Methods
10.8
Categorical Data and Nonparametric Methods
10.9
Categorical Data and Nonparametric Methods
11.1
Linear Statistical Models
11.2
Linear Statistical Models
11.3
Linear Statistical Models
11.4
Linear Statistical Models
11.5
Linear Statistical Models
11.6
Linear Statistical Models
11.7
Linear Statistical Models
11.8
Linear Statistical Models
11.9
Linear Statistical Models
12.1
Simulation
12.2
Simulation
12.3
Simulation
12.4
Simulation
12.5
Simulation
12.6
Simulation
12.7
Simulation
Textbook Solutions for Probability and Statistics
Chapter 5.9 Problem 7
Question
Suppose that the random variables X1,...,Xk are independent and that Xi has the Poisson distribution with mean i (i = 1, . . . , k). Show that for each fixed positive integer n, the conditional distribution of the random vector X = (X1,...,Xk), given that k i=1 Xi = n, is the multinomial distribution with parameters n and p = (p1,...,pk), where pi = i k j=1 j
Solution
The first step in solving 5.9 problem number 7 trying to solve the problem we have to refer to the textbook question: Suppose that the random variables X1,...,Xk are independent and that Xi has the Poisson distribution with mean i (i = 1, . . . , k). Show that for each fixed positive integer n, the conditional distribution of the random vector X = (X1,...,Xk), given that k i=1 Xi = n, is the multinomial distribution with parameters n and p = (p1,...,pk), where pi = i k j=1 j
From the textbook chapter Special Distributions you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Probability and Statistics 4
Author
Morris H. DeGroot, Mark J. Schervish
ISBN
9780321500465
Answer: Suppose that the random variables X1,...,Xk are
Chapter 5.9 textbook questions
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Chapter 5: Problem 2 Probability and Statistics 4
Suppose that F is a continuous c.d.f. on the real line, and let 1 and 2 be numbers such that F (1) = 0.3 and F (2) = 0.8. If 25 observations are selected at random from the distribution for which the c.d.f. is F, what is the probability that six of the observed values will be less than 1, 10 of the observed values will be between 1 and 2, and nine of the observed values will be greater than 2?
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Chapter 5: Problem 3 Probability and Statistics 4
If five balanced dice are rolled, what is the probability that the number 1 and the number 4 will appear the same number of times?
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Chapter 5: Problem 4 Probability and Statistics 4
Suppose that a die is loaded so that each of the numbers 1, 2, 3, 4, 5, and 6 has a different probability of appearing when the die is rolled. For i = 1,..., 6, let pi denote the probability that the number i will be obtained, and suppose that p1 = 0.11, p2 = 0.30, p3 = 0.22, p4 = 0.05, p5 = 0.25, and p6 = 0.07. Suppose also that the die is to be rolled 40 times. Let X1 denote the number of rolls for which an even number appears, and let X2 denote the number of rolls for which either the number 1 or the number 3 appears. Find the value of Pr(X1 = 20 and X2 = 15)
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Chapter 5: Problem 5 Probability and Statistics 4
Suppose that 16 percent of the students in a certain high school are freshmen, 14 percent are sophomores, 38 percent are juniors, and 32 percent are seniors. If 15 students are selected at random from the school, what is the probability that at least eight will be either freshmen or sophomores?
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Chapter 5: Problem 6 Probability and Statistics 4
In Exercise 5, let X3 denote the number of juniors in the random sample of 15 students, and let X4 denote the number of seniors in the sample. Find the value of E(X3 X4) and the value of Var(X3 X4).
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Chapter 5: Problem 7 Probability and Statistics 4
Suppose that the random variables X1,...,Xk are independent and that Xi has the Poisson distribution with mean i (i = 1, . . . , k). Show that for each fixed positive integer n, the conditional distribution of the random vector X = (X1,...,Xk), given that k i=1 Xi = n, is the multinomial distribution with parameters n and p = (p1,...,pk), where pi = i k j=1 j
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Chapter 5: Problem 8 Probability and Statistics 4
Suppose that the parts produced by a machine can have three different levels of functionality: working, impaired, defective. Let p1, p2, and p3 = 1 p1 p2 be the probabilities that a part is working, impaired, and defective, respectively. Suppose that the vector p = (p1, p2) is unknown but has a joint distribution with p.d.f. f (p1, p2) = 12p2 1 for 0 < p1, p2 < 1 and p1 + p2 < 1, 0 otherwise. Suppose that we observe 10 parts that are conditionally independent given p, and among those 10 parts, eight are working and two are impaired. Find the conditional p.d.f. of p given the observed parts. Hint: You might find Eq. (5.8.2) helpful.
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