Chapter 9 Bivariate Random Variables

Before examining the joint distribution of several random variables, focus first on the bivariate case. The aim is to investigate how two random variables change together through an analysis of their joint distribution.

We can derive the marginal distribution of each individual random variable from their joint distribution. If two random variables X and Y are defined on the same sample space, and we call their respective support sets A and B then the function that is the joint distribution is given by \(f(x,y)= \mathbb{P}(X=x, Y=y)\).

It is clear that for a bivariate distribution, that

\[ \int_{A}\int_{B}f(x,y)dxdy=1 \qquad \text{and} \qquad \sum_{x \in A}\sum_{y \in B}p(x,y)=1 \]

9.1 Independence and Correlation of Two RVs

When studying the relationship between two random variables, there are two primary characteristics we are interested in– namely, independence and correlation. Independence measures a relationship between X and Y, while correlation measures a linear relationship between X and Y.

9.1.1 Marginal distributions to determine independence

Two random variables are independent if and only if the product of their marginal distributions is equal to the joint distribution.

\[ f(x,y) = f(x) \cdot f(y) \]

9.1.2 Marginal expectation to determine correlation

Two random variables are uncorrelated if their covariance, or equivalently their correlation (the standardized covariance) is a value that is close to zero

\[ E[XY] = E[X] \cdot E[Y] \]

9.2 Correlation

Correlation is a measure of the strength of a linear relationship.

  • Two variables are not correlated if their correlation equals zero (covariance = 0)

  • if you have independence you know they are uncorrelated. but uncorrelated does not mean they are independent

9.3 Properties of Covariance

Derived from expected value, which is a linear operator.

\[ Cov(x,y) = E[XY] - E[X]E[Y] \]

  • independence is stronger than correlation

Additional notes:

  • In bivariate normal distribution, no correlation between variables actually implies independence.