6. We transmit a random variable X, which satisfies the distribution N(0, 1). The variable is corrupted in the channel by a N(0, 1) noise Y and becomes Z. In other words, Y~ N(0, 1), X and Y are independent, and Z = X + Y. (a) Find E[(X - aZ)²], and find a to minimize this expectation. (b) Find the conditional pdf fz|x(z|x). (c) Find the joint probability density function fx,z(x, z) = fz|x(z|x)ƒx(x). (d) Find the conditional probability density function fx|z(x|z). (e) Find E[X|Z].
6. We transmit a random variable X, which satisfies the distribution N(0, 1). The variable is corrupted in the channel by a N(0, 1) noise Y and becomes Z. In other words, Y~ N(0, 1), X and Y are independent, and Z = X + Y. (a) Find E[(X - aZ)²], and find a to minimize this expectation. (b) Find the conditional pdf fz|x(z|x). (c) Find the joint probability density function fx,z(x, z) = fz|x(z|x)ƒx(x). (d) Find the conditional probability density function fx|z(x|z). (e) Find E[X|Z].
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Problem 6: Analysis of a Corrupted Random Variable
We transmit a random variable \( X \), which satisfies the distribution \( \mathcal{N}(0, 1) \). The variable is corrupted in the channel by a \( \mathcal{N}(0, 1) \) noise \( Y \) and becomes \( Z \). In other words, \( Y \sim \mathcal{N}(0, 1) \), \( X \) and \( Y \) are independent, and \( Z = X + Y \).
#### (a) Find \( \mathbb{E}[(X - \alpha Z)^2] \), and find \( \alpha \) to minimize this expectation.
#### (b) Find the conditional pdf \( f_{Z|X}(z|x) \).
#### (c) Find the joint probability density function \( f_{X,Z}(x,z) = f_{Z|X}(z|x) f_X(x) \).
#### (d) Find the conditional probability density function \( f_{X|Z}(x|z) \).
#### (e) Find \( \mathbb{E}[X|Z] \).
---
### Solution Outline
1. **Objective**: Given the model where \( Z = X + Y \), with \( X \sim \mathcal{N}(0, 1) \) and \( Y \sim \mathcal{N}(0, 1) \), figure out various statistical properties and relationships between \( X \) and \( Z \).
2. **Key Concepts**:
- Probability Density Functions (PDFs)
- Expectation and Minimization of functions
- Conditional Distributions
- Joint Distributions
3. **Steps**:
- Calculate the expected value \( \mathbb{E}[(X - \alpha Z)^2] \).
- Derive the conditional probability density function \( f_{Z|X}(z|x) \).
- Use the above to find the joint PDF \( f_{X,Z}(x,z) \).
- From the joint PDF, derive the conditional PDF \( f_{X|Z}(x|z) \).
- Finally, find the conditional expectation \( \mathbb{E}[X|Z] \).
The problem touches on fundamental concepts in probability theory and will often appear in courses related to random](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe5b4bd-a3c0-40f8-aa61-fdbe86d977d3%2F9c1b8525-927b-4a89-b5ef-3273033b02a0%2F08ri51h_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 6: Analysis of a Corrupted Random Variable
We transmit a random variable \( X \), which satisfies the distribution \( \mathcal{N}(0, 1) \). The variable is corrupted in the channel by a \( \mathcal{N}(0, 1) \) noise \( Y \) and becomes \( Z \). In other words, \( Y \sim \mathcal{N}(0, 1) \), \( X \) and \( Y \) are independent, and \( Z = X + Y \).
#### (a) Find \( \mathbb{E}[(X - \alpha Z)^2] \), and find \( \alpha \) to minimize this expectation.
#### (b) Find the conditional pdf \( f_{Z|X}(z|x) \).
#### (c) Find the joint probability density function \( f_{X,Z}(x,z) = f_{Z|X}(z|x) f_X(x) \).
#### (d) Find the conditional probability density function \( f_{X|Z}(x|z) \).
#### (e) Find \( \mathbb{E}[X|Z] \).
---
### Solution Outline
1. **Objective**: Given the model where \( Z = X + Y \), with \( X \sim \mathcal{N}(0, 1) \) and \( Y \sim \mathcal{N}(0, 1) \), figure out various statistical properties and relationships between \( X \) and \( Z \).
2. **Key Concepts**:
- Probability Density Functions (PDFs)
- Expectation and Minimization of functions
- Conditional Distributions
- Joint Distributions
3. **Steps**:
- Calculate the expected value \( \mathbb{E}[(X - \alpha Z)^2] \).
- Derive the conditional probability density function \( f_{Z|X}(z|x) \).
- Use the above to find the joint PDF \( f_{X,Z}(x,z) \).
- From the joint PDF, derive the conditional PDF \( f_{X|Z}(x|z) \).
- Finally, find the conditional expectation \( \mathbb{E}[X|Z] \).
The problem touches on fundamental concepts in probability theory and will often appear in courses related to random
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