Introduction

The Gaussian distribution, also known as the normal distribution, is one of the most widely used probability distributions in statistics and machine learning. It is characterized by its bell-shaped curve and is commonly used to model continuous random variables. In this blog, we will explore the concept of the conditional distribution of a Gaussian distribution, explain the relevant formulas and equations, and provide examples to enhance understanding.

The Gaussian Distribution

The Gaussian distribution is defined by two parameters: the mean (μ) and the variance (σ^2). The probability density function (pdf) of a univariate Gaussian distribution is given by:

f(x; μ, σ^2) = (1 / sqrt(2πσ^2)) * exp(-(x - μ)^2 / (2σ^2))

Where:

• f(x; μ, σ^2) represents the probability density function of the Gaussian distribution at a specific value x.
• μ is the mean of the distribution.
• σ^2 is the variance of the distribution.

The Conditional Distribution

The conditional distribution refers to the distribution of a random variable given the knowledge or information about another random variable. In the case of the Gaussian distribution, the conditional distribution can be derived using the concept of conditional probability.

The conditional distribution of a Gaussian random variable Y, given the value of another random variable X, is denoted as Y | X. It represents the distribution of Y when X is fixed or known.

Formula for Conditional Distribution of Gaussian Distribution:

The conditional distribution of a Gaussian distribution is itself a Gaussian distribution. The conditional mean (μ_Y|X) and conditional variance (σ^2_Y|X) can be calculated using the following formulas:

μ_Y|X = μ_Y + ρ * (σ_Y / σ_X) * (X - μ_X)

σ^2_Y|X = σ^2_Y * (1 - ρ^2)

Where:

• μ_Y and σ^2_Y are the mean and variance of the original Gaussian distribution of Y.
• μ_X and σ^2_X are the mean and variance of the original Gaussian distribution of X.
• ρ represents the correlation coefficient between X and Y.

Example: Conditional Distribution of Gaussian Distribution

Let's consider an example to better understand the concept of the conditional distribution of a Gaussian distribution.

Suppose we have two random variables, X and Y, following a bivariate Gaussian distribution. The joint distribution is given by:

f(X, Y; μ_X, μ_Y, σ^2_X, σ^2_Y, ρ) = (1 / (2πσ_Xσ_Ysqrt(1 - ρ^2))) * exp(-[1 / (2(1 - ρ^2))][(X - μ_X)^2 / σ^2_X - 2ρ(X - μ_X)(Y - μ_Y) / (σ_X*σ_Y) + (Y - μ_Y)^2 / σ^2_Y])

Suppose we want to find the conditional distribution of Y given the value of X, i.e., Y|X=x.

Using the formulas for the conditional mean and variance, we can calculate the parameters of the conditional distribution:

μ_Y|X = μ_Y + ρ * (σ_Y / σ_X) * (x - μ_X)

σ^2_Y|X = σ^2_Y * (1 - ρ^2)

This gives us the parameters for the conditional Gaussian distribution of Y given X.

Conclusion

The conditional distribution of a Gaussian distribution provides valuable insights into the distribution of a random variable when another random variable is fixed or known. By calculating the conditional mean and variance using the appropriate formulas, we can obtain the parameters of the conditional Gaussian distribution. This allows us to understand how the distribution of one variable changes based on the value or information of another variable.