The bivariate normal distribution is a probability distribution that describes the joint distribution of two continuous random variables. It is an extension of the univariate normal distribution, which describes the distribution of a single random variable.
In the bivariate normal distribution, each variable follows a normal distribution individually, and their joint distribution is determined by their means, variances, and the correlation between the variables. The distribution is defined by four parameters: the means of each variable (μ₁, μ₂), the variances of each variable (σ₁², σ₂²), and the correlation coefficient (ρ), which measures the linear dependence between the variables.
The bivariate normal distribution is characterized by its bell-shaped, symmetric density function. The shape of the distribution is influenced by the means, variances, and the correlation between the variables. When the two variables are perfectly linearly dependent (correlation coefficient = ±1), the distribution takes an elliptical form along the line of dependence. As the correlation decreases, the elliptical shape becomes more elongated or flattened.
The bivariate normal distribution has several important properties, such as the marginal distributions of each variable being normal, and the conditional distributions given one variable being normal as well. Additionally, any linear combination of the variables that satisfies certain conditions is also normally distributed.
The bivariate normal distribution has numerous applications in statistics, econometrics, finance, and other fields where the joint distribution of two variables needs to be understood and analyzed.
