Joint Continuous Random Variables

A continuous random variable can take on an uncountable number of values within a certain range.

Example

Examples include height, weight, time, temperature, and any measurement that can take on a wide range of real numbers.

Joint Continuous random variables deal with the simultaneous behavior of two or more continuous random variables.

Example

Consider two continuous random variables, \(X\) and \(Y\). \(X\) represents the time it takes for a customer to arrive at a store, and \(Y\) represents the amount of money the customer spends during their visit. Both \(X\) and \(Y\) can take on a wide range of real-number values.

Joint Probability Density Function (PDF)

When dealing with two continuous random variables, \(X\) and \(Y\), the joint PDF \(f_{XY}(x,y)\) defines the probability density for every pair of values \((x,y)\) that \(X\) and \(Y\) can take simultaneously.
This function represents how likely it is for \(X\) and \(Y\) to jointly fall within a specific region in their respective value spaces.
The integral of this joint PDF over a region of the XY plane gives the probability that (X, Y) falls within that region.

Properties of PDF

A function \(f(x,y)\) is said to be a joint density function if

\(f(x,y) \geq 0\) , i.e. \(f\) is non-negative
\(\int \int^{\infty}_{ - \infty} f(x,y)dxdy = 1\)
\(f(x,y)\) is piecewise continuous in each random variable.

What does Piecewise Continuous mean?

"Piecewise continuous" is a term used to describe a function that is continuous except for a finite number of isolated points or intervals where it may have discontinuities. In other words, a piecewise continuous function can be thought of as a function that is continuous over certain subintervals of its domain but may have jumps or discontinuities at specific points or subintervals.

Example

Let \(X\) and \(Y\) have joint density

\[ f_{XY} = \begin{cases} 1 & 0 < x < 1 , 0 < y < 1 \\ 0 & otherwise \end{cases} \]

This specific example is also known as uniform unit square.

In the context of the arrival time and spending example, the joint PDF \(f_{XY}(x,y)\) might represent the probability density of a customer arriving at time \(x\) and spending \(y\) dollars. It provides the likelihood of simultaneous occurrences of these two events.

2D Uniform Distribution

For some (reasonable) region \(D\) in \(\mathbb{R}^2\) with total area \(|D|\).

We say that \((X,Y) \sim D\) if they have the joint density

\[ f_{XY} = \begin{cases} \frac{1}{|D|} & (x,y) \in D \\ 0 & otherwise \end{cases} \]

For any sub region \(A\) of \(D\) , \(P((X,Y) \in A) = \frac{|A|}{|D|} = \frac{\text{Area of A}}{\text{Area of D}}\)

Marginal Density

Just as with single random variables, you can find the marginal PDFs for X and Y from the joint PDF.
The marginal PDF for \(X\) , denoted as \(f_X(x)\) , is obtained by integrating the joint PDF \(f_{XY}(x,y)\) with respect to \(Y\) over the entire range of \(Y\) values.
Similarly , the marginal PDF for \(Y\) , denoted as \(f_Y(y)\) , is obtained by integrating the joint PDF \(f_{XY}(x,y)\) with respect to \(X\) over the entire range of \(X\) values.

Suppose \((X,Y)\) have joint density \(f_{XY}(x,y).\) Then,

\(X\) has the marginal density \(f_X(x)\) = \(\int^{\infty}_{y = - \infty} f_{XY}(x,y)dy\)
\(Y\) has the marginal density \(f_Y(y)\) = \(\int^{\infty}_{x = - \infty} f_{XY}(x,y)dy\)

Example

From the joint PDF \(f_{XY}(x,y)\) , you can find the marginal PDF for \(X\) (\(f_X(x)\)) and \(Y\) (\(f_Y(y)\)) separately. \(f_X(x)\) would represent the distribution of arrival times , and \(f_Y(y)\) would represent the distribution of spending amounts

Independence of Random Variables

Independence means that the behavior of one random variable does not influence the behavior of the other.

\((X,Y)\) with joint density \(f_{XY}(x,y)\) are independent if

\[ f_{XY}(x,y) = f_X(x)f_Y(y) \]

where \(f_X(x)\) and \(f_Y(y)\) are marginal densities of \(X\) and \(Y\) respectively.

Example

Suppose \(X\) represents the temperature in one city, and \(Y\) represents the temperature in another city. If these cities are far apart and have independent weather patterns, then \(X\) and \(Y\) may be considered independent.

Conditional Density

Conditional Density describe the probability of one random variable given the other.

Let \((X,Y)\) be random variables with joint density \(f_{XY}(x,y)\). Let \(f_X(x)\) and \(f_Y(y)\) be the marginal densities.

For \(a\) such that \(f_X(a) > 0\) , the conditional density of \(Y\) given \(X=a\) , denoted \(f_{Y|X=a}(y)\) , is defined as

\[ f_{Y|X=a} = \frac{f_{XY}(a,y)}{f_X(a)} \]

For \(b\) such that \(f_Y(b) > 0\) , the conditional density of \(X\) given \(Y=b\) , denoted \(f_{X|Y=b}(x)\) , is defined as

\[ f_{X|Y=b}(x) = \frac{f_{XY}(x,b)}{f_Y(b)} \]

Properties of Conditional Density

Both the conditional densities are valid densities in one dimension. So , the "conditional" random variables \((Y|X=a)\) and \((X|Y=b)\) are well defined.
\(\text{Joint} = \text{Marginal} \times \text{Conditional}\) , for \(x=a\) and \(y=b\) such that \(f_X(a) > 0\) and \(f_Y(b) > 0\)

\[ f_{XY}(a,b) = f_X(a)f_{Y|X=a}(b) = f_Y(b)f_{X|Y=b}(a) \]

Example

Continuing with the temperature example, you might want to find \(f_{X|Y}(x|y)\), which would represent the PDF of temperature in one city given a specific temperature in the other city. This could help predict temperature changes in one city based on the observed temperature in the other city.