Bivariate Discrete Cumulative Distribution Function

Data Science and A.I. Lecture Series

Author: Bindeshwar Singh Kushwaha

Institute: PostNetwork Academy

Joint and Marginal Distribution Functions for Discrete Random Variables

Two-Dimensional Joint Distribution Function

The distribution function of the two-dimensional random variable \((X, Y)\) for all real \(x\) and \(y\) is defined as:

\[
F(x,y) = P(X \leq x, Y \leq y)
\]

Joint Probability Table

\[
\begin{array}{c|cc}
X \backslash Y & 1 & 2 \\
\hline
1 & 0.1 & 0.2 \\
2 & 0.1 & 0.3 \\
3 & 0.2 & 0.1 \\
\end{array}
\]

Example 1: Joint and Marginal Distribution

\[
\begin{array}{c|cc}
X \backslash Y & 1 & 2 \\
\hline
1 & 0.1 & 0.2 \\
2 & 0.1 & 0.3 \\
3 & 0.2 & 0.1 \\
\end{array}
\]

\[
F(2,2) = P(X \leq 2, Y \leq 2)
\]

\[
= P(X=2,Y=2) + P(X=1,Y=2) + P(X=2,Y=1) + P(X=1,Y=1)
\]

\[
= 0.3 + 0.1 + 0.2 + 0.1 = 0.7
\]

\[
F(3,2) = P(X \leq 3, Y \leq 2)
\]

\[
= F(2,2) + P(X=3,Y=2) + P(X=3,Y=1)
\]

\[
= 0.7 + 0.1 + 0.2 = 1
\]

\[
F_X(3) = P(X \leq 3) = P(X=1) + P(X=2) + P(X=3)
\]

\[
= (0.1+0.2) + (0.1+0.3) + (0.2+0.1) = 1
\]

Example 2: Joint Probability Distribution

\[
\begin{array}{c|cc}
X \backslash Y & 0 & 1 \\
\hline
0 & 2/9 & 1/9 \\
1 & 1/9 & 5/9 \\
\end{array}
\]

\[
F(0,0) = P(X=0,Y=0) = \frac{2}{9}
\]

\[
F(0,1) = P(X=0,Y=0) + P(X=0,Y=1)
\]

\[
= \frac{2}{9} + \frac{1}{9} = \frac{3}{9}
\]

\[
F(1,0) = P(X=0,Y=0) + P(X=1,Y=0)
\]

\[
= \frac{2}{9} + \frac{1}{9} = \frac{3}{9}
\]

\[
F(1,1) = P(X=0,Y=0) + P(X=0,Y=1) + P(X=1,Y=0) + P(X=1,Y=1)
\]

\[
= \frac{2}{9} + \frac{1}{9} + \frac{1}{9} + \frac{5}{9} = 1
\]

Video

PDF

BivariateCDFDiscrete

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