Comments about "Complex random variable" in Wikipedia

This document contains comments about the article Complex random variable in Wikipedia

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In the last paragraph I explain my own opinion.

Contents

• 1 Definition
• 2 Examples
• 2.1 Simple example
• 2.2 Uniform distribution
• 2.3 Complex normal distribution
• 3 Cumulative distribution function
• 4 Probability density function
• 5 Expectation
• 6 Variance and pseudo-variance
• 6.1 Pseudo-variance
• 7 Covariance matrix of real and imaginary parts
• 8 Covariance and pseudo-covariance
• 9 Circular symmetry
• 10 Proper complex random variables
• 11 Cauchy-Schwarz inequality
• 12 Characteristic function
• 13. See also

Reflection

• Reflection 1 -
• Reflection 2 -
• Reflection 3 -

Introduction

The article starts with the following sentence.

1 Definition

2 Examples

2.1 Simple example

Probability P(z) Value z 1/4 1+i 1/4 1-i 1/2 2

Consider a random variable that may take only the three complex values 1 + i , 1 - i , 2 with probabilities as specified in the table. This is a simple example of a complex random variable. The expectation of this random variable may be simply calculated: E[Z] = 1/4(1+i) + 1/4 (1-i) =1/2*2 = 3/2

This works because the probabilities in line 1 and 2 are identical. If not the calculated expectation value would be a complex number, and what would that mean?

2.2 Uniform distribution

Another example of a complex random variable is the uniform distribution over the filled unit circle, i.e. the set { z inclusif C | |z| = 1 } This random variable is an example of a complex random variable for which the probability density function is defined.

Tricky to compare within the real world

2.3 Complex normal distribution

3 Cumulative distribution function

4 Probability density function

5 Expectation

6 Variance and pseudo-variance

6.1 Pseudo-variance

7 Covariance matrix of real and imaginary parts

8 Covariance and pseudo-covariance

9 Circular symmetry

10 Proper complex random variables

11 Cauchy-Schwarz inequality

12 Characteristic function

13. See also

Following is a list with "Comments in Wikipedia" about related subjects

• Probability density function Comments

Reflection 1

Reflection 2

Reflection 3

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