Let X and Y be discrete random variables that take the values 1,2, ... N with equal probability, where N is a positive integer. X and Y are independent, so in particular P(X ≤ a and Y ≤ b) = P(X ≤ a)P (Y < b). Let Z max (X, Y). Note that Z is also a random variable that takes values from 1 to N. a. Write down the probability distribution function for X (which is also the probability distribution function for Y). b. Find the expected value of X c. Find P(Z < 20) (Hint: Your answer will involve the parameter N. There are two cases: what are they? Note also that max (X, Y) < 20 if and only if both X < 20 and Y ≤ 20). d. Find the cdf of Z. (Hint: Write down the definition of Fz(x).) e. Using the cdf from part d., find the probability distribution function for Z. (NOTE I am being very specific here. You MUST use the cdf of to do this problem. If you do it a different way, you receive no credit.) f. Find E[Z]. (Hint: you will need the formulas for EN-1n and EN_₁n²) E[Z] g. Find limx→00 and comment on your answer. Can you think of a practical situation E[X]' in which this information might be useful?
Let X and Y be discrete random variables that take the values 1,2, ... N with equal probability, where N is a positive integer. X and Y are independent, so in particular P(X ≤ a and Y ≤ b) = P(X ≤ a)P (Y < b). Let Z max (X, Y). Note that Z is also a random variable that takes values from 1 to N. a. Write down the probability distribution function for X (which is also the probability distribution function for Y). b. Find the expected value of X c. Find P(Z < 20) (Hint: Your answer will involve the parameter N. There are two cases: what are they? Note also that max (X, Y) < 20 if and only if both X < 20 and Y ≤ 20). d. Find the cdf of Z. (Hint: Write down the definition of Fz(x).) e. Using the cdf from part d., find the probability distribution function for Z. (NOTE I am being very specific here. You MUST use the cdf of to do this problem. If you do it a different way, you receive no credit.) f. Find E[Z]. (Hint: you will need the formulas for EN-1n and EN_₁n²) E[Z] g. Find limx→00 and comment on your answer. Can you think of a practical situation E[X]' in which this information might be useful?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Question
Please show your work
![Let \( X \) and \( Y \) be discrete random variables that take the values \( 1, 2, \ldots, N \) with equal probability, where \( N \) is a positive integer. \( X \) and \( Y \) are independent, so in particular \( P(X \leq a \text{ and } Y \leq b) = P(X \leq a)P(Y \leq b) \). Let \( Z = \max(X, Y) \). Note that \( Z \) is also a random variable that takes values from 1 to \( N \).
a. Write down the probability distribution function for \( X \) (which is also the probability distribution function for \( Y \)).
b. Find the expected value of \( X \).
c. Find \( P(Z \leq 20) \).
(Hint: Your answer will involve the parameter \( N \). There are two cases: what are they? Note also that \( \max(X, Y) \leq 20 \) if and only if both \( X \leq 20 \) and \( Y \leq 20 \)).
d. Find the cdf of \( Z \). (Hint: Write down the definition of \( F_Z(x) \)).
e. Using the cdf from part d., find the probability distribution function for \( Z \). (NOTE: I am being very specific here. You MUST use the cdf to do this problem. If you do it a different way, you receive no credit.)
f. Find \( E[Z] \). (Hint: you will need the formulas for \( \sum_{n=1}^{N} n \) and \( \sum_{n=1}^{N} n^2 \))
g. Find \( \lim_{N \to \infty} \frac{E[Z]}{E[X]} \), and comment on your answer. Can you think of a practical situation in which this information might be useful?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F5bc8aed1-40de-4362-8b9d-8116949e3175%2F5habi6o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let \( X \) and \( Y \) be discrete random variables that take the values \( 1, 2, \ldots, N \) with equal probability, where \( N \) is a positive integer. \( X \) and \( Y \) are independent, so in particular \( P(X \leq a \text{ and } Y \leq b) = P(X \leq a)P(Y \leq b) \). Let \( Z = \max(X, Y) \). Note that \( Z \) is also a random variable that takes values from 1 to \( N \).
a. Write down the probability distribution function for \( X \) (which is also the probability distribution function for \( Y \)).
b. Find the expected value of \( X \).
c. Find \( P(Z \leq 20) \).
(Hint: Your answer will involve the parameter \( N \). There are two cases: what are they? Note also that \( \max(X, Y) \leq 20 \) if and only if both \( X \leq 20 \) and \( Y \leq 20 \)).
d. Find the cdf of \( Z \). (Hint: Write down the definition of \( F_Z(x) \)).
e. Using the cdf from part d., find the probability distribution function for \( Z \). (NOTE: I am being very specific here. You MUST use the cdf to do this problem. If you do it a different way, you receive no credit.)
f. Find \( E[Z] \). (Hint: you will need the formulas for \( \sum_{n=1}^{N} n \) and \( \sum_{n=1}^{N} n^2 \))
g. Find \( \lim_{N \to \infty} \frac{E[Z]}{E[X]} \), and comment on your answer. Can you think of a practical situation in which this information might be useful?
Expert Solution
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 3 steps with 6 images
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON