Lecture 7: Joint Distribution
STA237: Probability, Statistics, and Data Analysis I
PhD Student, DoSS, University of Toronto
June 6, 2022
For example, the probability of selling 5 cups of coffee and 5 muffins on a day would be indicated by …
How about the probability of selling exactly 5 cups of coffee on a day?
How about the probability of selling less than 5 cups of coffee and less than 4 muffins on a day?
This is an example of a joint distribution of two discrete random variables.
The two random variables arise from the same sample space and the joint distribution describe probabilities of all possible pairs of their values.
Joint probability density function
Recall …
- A probability density function is a non-negative function
- To compute a probability, we need its integral - that is, the area under the curve
- Total area under the curve is 1
Extending to a pair …
- A joint probability density function is a non-negative function in 2 dimensions
- To compute a probability, we need to compute double integrals - that is, the volume under the surface
- Total volume under the surface is 1
Example: 2.7.8 from Evans and Rosenthal
Suppose \(X\) and \(Y\) have a joint continuous distribution with joint density
\[f_{X,Y}\left(x,y\right)=\begin{cases}120\cdot x^3\cdot y & x\ge 0, y\ge 0, x+y\le1 \\ 0 &\text{otherwise.}\end{cases}\]
- Check \(f\left(x,y\right) \ge 0\) for all values of \(x\) and \(y\)
- Check \(\int_{-\infty}^\infty\int_{-\infty}^\infty f\left(x,y\right) dxdy = 1\)
- Compute \(F_{X,Y}(0.5, 0.5)\) and \(P(X\le 0.5)\).
Example: 2.7.8 from Evans and Rosenthal
\[F_{X,Y}(0.5, 0.5)\]
- \(p_{X,Y}(0.5, 0.5)=P(X\le 0.5, Y\le 0.5)\)
- \(=\int_{-\infty}^{1/2}\int_{-\infty}^{1/2}f(x,y) dxdy\)
- \(=\int_0^{1/2}\int_0^{1/2}120\cdot x^3\cdot y dxdy\)
- \(=\int_0^{1/2} 120 \cdot (1/2)^4 / 4 \cdot y dy\)
- \(=30 \cdot {1/2}^4 \cdot (1/2)^2 / 2\)
- \(=15/64\)
Example: 2.7.8 from Evans and Rosenthal
\[P(X\le 0.5)\]
- \(P(X\le 0.5)\)
- \(=\int_{-\infty}^{???}\int_{-\infty}^{1/2}f(x,y) dxdy\)
- \(=\int_{-\infty}^{\infty}\int_{-\infty}^{1/2}f(x,y) dxdy\)
\[\neq\int_0^{1}\int_{0}^{1/2} 120 \cdot x^3 \cdot y dxdy\]
\(f(x,y)\) is not \(120\cdot x^3 \cdot y\) for all values of \(y\) from 0 to 1 when \(x\) is between 0 and 1/2.
Example: 2.7.8 from Evans and Rosenthal
\[P(X\le 0.5)\]
- \(P(X\le 0.5)\)
- \(=\int_{-\infty}^{???}\int_{-\infty}^{1/2}f(x,y) dxdy\)
- \(=\int_{-\infty}^{\infty}\int_{-\infty}^{1/2}f(x,y) dxdy\)
\[=\int_{0}^{1/2}\int_0^{1-x} 120 \cdot x^3 \cdot y dydx\]
We can switch the order of integrals when working with probability density functions and evaluate \(y\) in terms of \(x\).