Suppose that Alejandro is planting a garden of tulips. Let X be the Bernoulli random variable that returns 1 if the tulip is red, and 0 otherwise. Suppose Y is another Bernoulli random variable, independent of X, that returns 1 if the tulip survives longer than 30 days, and 0 otherwise. Both X and Y have parameters 1/2. Let Z be the random variable that returns the remainder of the division of X+Y by 2. For example, if X 1 and Y = 0, Z = remainder()= 1 (a) Prove that Z is also a Bernoulli random variable, also with parameter 1/2. (b) Prove that X, Y, Z are pairwise independent but not mutually independent. (c) By computing Var[X+Y+Z] according to the alternative formula for variance and using the variance of Bernoulli r.v.'s, verify that Var[X+Y + Z] = Var[X] +Var[Y] +Var[Z] (observe that this also follows from the proposition on slide 5 of the lecture segment entitled "Binomial distribution").
Suppose that Alejandro is planting a garden of tulips. Let X be the Bernoulli random variable that returns 1 if the tulip is red, and 0 otherwise. Suppose Y is another Bernoulli random variable, independent of X, that returns 1 if the tulip survives longer than 30 days, and 0 otherwise. Both X and Y have parameters 1/2. Let Z be the random variable that returns the remainder of the division of X+Y by 2. For example, if X 1 and Y = 0, Z = remainder()= 1 (a) Prove that Z is also a Bernoulli random variable, also with parameter 1/2. (b) Prove that X, Y, Z are pairwise independent but not mutually independent. (c) By computing Var[X+Y+Z] according to the alternative formula for variance and using the variance of Bernoulli r.v.'s, verify that Var[X+Y + Z] = Var[X] +Var[Y] +Var[Z] (observe that this also follows from the proposition on slide 5 of the lecture segment entitled "Binomial distribution").
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
Related questions
Question
urgently needed
![4
Suppose that Alejandro is planting a garden of tulips. Let X be the Bernoulli random
variable that returns 1 if the tulip is red, and 0 otherwise. Suppose Y is another Bernoulli
random variable, independent of X, that returns 1 if the tulip survives longer than 30 days,
and 0 otherwise. Both X and Y have parameters 1/2. Let Z be the random variable that
returns the remainder of the division of X +Y by 2. For example, if X = 1 and Y = 0,
Z = remainder() = 1
(a) Prove that Z is also a Bernoulli random variable, also with parameter 1/2.
(b) Prove that X, Y, Z are pairwise independent but not mutually independent.
(c) By computing Var[X+Y+Z] according to the alternative formula for variance and using
the variance of Bernoulli r.v.'s, verify that Var[X+Y+Z] = Var[X]+Var[Y] +Var[Z]
%3D
(observe that this also follows from the proposition on slide 5 of the lecture segment entitled
"Binomial distribution").](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e8c00cd-a7dc-48ce-8fd9-4b4894d2734e%2F770dfc86-2958-46be-a729-c2c77bed5088%2Fxd2rjml_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4
Suppose that Alejandro is planting a garden of tulips. Let X be the Bernoulli random
variable that returns 1 if the tulip is red, and 0 otherwise. Suppose Y is another Bernoulli
random variable, independent of X, that returns 1 if the tulip survives longer than 30 days,
and 0 otherwise. Both X and Y have parameters 1/2. Let Z be the random variable that
returns the remainder of the division of X +Y by 2. For example, if X = 1 and Y = 0,
Z = remainder() = 1
(a) Prove that Z is also a Bernoulli random variable, also with parameter 1/2.
(b) Prove that X, Y, Z are pairwise independent but not mutually independent.
(c) By computing Var[X+Y+Z] according to the alternative formula for variance and using
the variance of Bernoulli r.v.'s, verify that Var[X+Y+Z] = Var[X]+Var[Y] +Var[Z]
%3D
(observe that this also follows from the proposition on slide 5 of the lecture segment entitled
"Binomial distribution").
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Holt Mcdougal Larson Pre-algebra: Student Edition…](https://www.bartleby.com/isbn_cover_images/9780547587776/9780547587776_smallCoverImage.jpg)
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781305115545/9781305115545_smallCoverImage.gif)
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Holt Mcdougal Larson Pre-algebra: Student Edition…](https://www.bartleby.com/isbn_cover_images/9780547587776/9780547587776_smallCoverImage.jpg)
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781305115545/9781305115545_smallCoverImage.gif)
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning