Homework Help is Here – Start Your Trial Now!
Math
Probability
In this problem, we will consider a sequence of 12 independent Bernoulli random variables denoted by (X1, X2, X3, X₁, X5, X6, X7, X8, X9, X10, X11, X12). Each of the Bernoulli random variables takes the value 1 with probability 3/4 and takes the value 0 with probability 1/4. That is, for each i=1,2,..., 12, the random variable X, has the pmf (3/4, k=1, px, (k)= [1/4, k = 0. As mentioned above, all 12 of the random variables are independent of one another. (a) What is the expected number of 1's in the sequence (X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12)? (b) What is the variance of the number of 1's in the sequence? (c) Which of the following sequences has a higher probability of occurring? Explain. sequence a: sequence b: (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) (1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1,0) (d) Conditioned on the sequence having at least ten 1's, what is the probability that the sequence contains exactly eleven 1's? (e) Conditioned on the sequence having at least ten 1's, what is the expected number of 1's in the sequence? (f) Conditioned on the sequence having exactly six 1's, what is the probability that the first three numbers in the sequence are 1?

In this problem, we will consider a sequence of 12 independent Bernoulli random variables denoted by (X1, X2, X3, X₁, X5, X6, X7, X8, X9, X10, X11, X12). Each of the Bernoulli random variables takes the value 1 with probability 3/4 and takes the value 0 with probability 1/4. That is, for each i=1,2,..., 12, the random variable X, has the pmf (3/4, k=1, px, (k)= [1/4, k = 0. As mentioned above, all 12 of the random variables are independent of one another. (a) What is the expected number of 1's in the sequence (X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12)? (b) What is the variance of the number of 1's in the sequence? (c) Which of the following sequences has a higher probability of occurring? Explain. sequence a: sequence b: (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) (1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1,0) (d) Conditioned on the sequence having at least ten 1's, what is the probability that the sequence contains exactly eleven 1's? (e) Conditioned on the sequence having at least ten 1's, what is the expected number of 1's in the sequence? (f) Conditioned on the sequence having exactly six 1's, what is the probability that the first three numbers in the sequence are 1?

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section9.3: Binomial Probability
Problem 2E: If a binomial experiment has probability p success, then the probability of failure is...
See similar textbooks
icon
Related questions
Question
100%

Please help with answering this question

In this problem, we will consider a sequence of 12 independent Bernoulli random variables denoted by (X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11 X12). Each of the Bernoulli random variables takes the value 1 with probability 3/4 and takes the value 0 with probability 1/4. That is, for each i = 1,2,..., 12, the random variable X, has the pmf px, (k) = [3/4, k=1, 1/4, k=0. As mentioned above, all 12 of the random variables are independent of one another. (a) What is the expected number of 1's in the sequence (X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12)? (b) What is the variance of the number of 1's in the sequence? (c) Which of the following sequences has a higher probability of occurring? Explain. sequence a: sequence b: (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) (1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1,0) (d) Conditioned on the sequence having at least ten 1's, what is the probability that the sequence contains exactly eleven 1's? (e) Conditioned on the sequence having at least ten 1's, what is the expected number of 1's in the sequence? (f) Conditioned on the sequence having exactly six 1's, what is the probability that the first three numbers in the sequence are 1?
Transcribed Image Text:In this problem, we will consider a sequence of 12 independent Bernoulli random variables denoted by (X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11 X12). Each of the Bernoulli random variables takes the value 1 with probability 3/4 and takes the value 0 with probability 1/4. That is, for each i = 1,2,..., 12, the random variable X, has the pmf px, (k) = [3/4, k=1, 1/4, k=0. As mentioned above, all 12 of the random variables are independent of one another. (a) What is the expected number of 1's in the sequence (X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12)? (b) What is the variance of the number of 1's in the sequence? (c) Which of the following sequences has a higher probability of occurring? Explain. sequence a: sequence b: (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) (1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1,0) (d) Conditioned on the sequence having at least ten 1's, what is the probability that the sequence contains exactly eleven 1's? (e) Conditioned on the sequence having at least ten 1's, what is the expected number of 1's in the sequence? (f) Conditioned on the sequence having exactly six 1's, what is the probability that the first three numbers in the sequence are 1?
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 8 steps with 15 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
College Algebra
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
SEE MORE TEXTBOOKS