1. Show for RVs (X, Y), independence results in px,y = 0. P
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Answer 1,2,3 please
![1. Show for RVs (X, Y), independence results in px,y = 0.
2. For jointly Gaussian RVs, show px,y = 0 results in independence.
3. Let X~ N(0, 1). Let Y = 1 with probability and Y = -X with probability (e.g., depending on
throw of a coin head or tail, we get Y = X or Y = -X).
Show Y~ N(0, 1). Hint: Find the CDF of Y and note that
P[Y <a] = P[Y <a/A] P[A] + P[Y <a Aº] P[A]
. Show px, y = 0
Show that X, Y are not independent.
• Find joint probability distribution of X, Y (simplify as much as you can).
Moral of the story: If two normal RVs are not jointly Gaussian, p = 0 does not lead to
independence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31398ba9-6632-4952-b9f2-59866c70af5c%2Fc794010e-e2a1-4e06-a550-4dbcdb9a7564%2Fy45xftd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Show for RVs (X, Y), independence results in px,y = 0.
2. For jointly Gaussian RVs, show px,y = 0 results in independence.
3. Let X~ N(0, 1). Let Y = 1 with probability and Y = -X with probability (e.g., depending on
throw of a coin head or tail, we get Y = X or Y = -X).
Show Y~ N(0, 1). Hint: Find the CDF of Y and note that
P[Y <a] = P[Y <a/A] P[A] + P[Y <a Aº] P[A]
. Show px, y = 0
Show that X, Y are not independent.
• Find joint probability distribution of X, Y (simplify as much as you can).
Moral of the story: If two normal RVs are not jointly Gaussian, p = 0 does not lead to
independence.
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