Joint Probability Density Function
The joint probability density function \(f(x,y)\) to handle simultaneous calculations of random variables \(X\) and \(Y\) can be expressed as
\[P({\color{blue}a_{1} < X < a_{2}}, {\color{red}b_{1} < Y < b_{2}}) = \displaystyle{\color{blue}\int_{a_{1}}^{a_{2}}}{\color{red}\int_{b_{1}}^{b_{2}}} \! {\color{purple}f(x,y)} \, {\color{red}dy} \, {\color{blue}dx}\]
- Each probability is between zero and one inclusively \[0 \leq {\color{purple}f(x,y)} \leq 1 \text{ for all } {\color{blue}$x$}, {\color{red}$y$}\]
- All probabilities add up to 100 percent \[{\color{red}\displaystyle\int_{-\infty}^{\infty}}{\color{blue}\displaystyle\int_{-\infty}^{\infty}} \! {\color{purple}f(x,y)} \, {\color{blue}dx} \, {\color{red}dy} = 1\]
Setting
For the examples in this lecture session, we will model the queues at In-n-Out with random variables
- \(X\): wait time to order food
- \(Y\): wait time to receive food
and a function of the form
\[{\color{purple}f(x,y)} = k{\color{blue}x}{\color{red}y}{\color{blue}e^{-x}}{\color{red}e^{-y/5}},\]
\[{\color{blue}x > 0}, \quad {\color{red}y > 0}\]
Normalization
Find the value of \(k\) so that \(f\) is a probability density function.
\[{\color{purple}f(x,y)} = k{\color{blue}x}{\color{red}y}{\color{blue}e^{-x}}{\color{red}e^{-y/5}}, \quad {\color{blue}x > 0}, \quad {\color{red}y > 0}\]
Joint Probability
\[{\color{purple}f(x,y)} = \displaystyle\frac{1}{25}{\color{blue}x}{\color{red}y}{\color{blue}e^{-x}}{\color{red}e^{-y/5}}, \quad {\color{blue}x > 0}, \quad {\color{red}y > 0}\]
Compute the probability that you will take {between 1 and 2 minutes to order} and wait {between 3 and 4 minutes to receive} your food.
Joint Cumulative Distribution Function
In general, we handle the probability calculations with the joint cumulative distribution function \(F(a,b)\)
% joint cumulative distribution function \[{\color{purple}F(a,b)} = P({\color{blue}X \leq a}, {\color{red}Y \leq b}) = {\color{blue}\displaystyle\int_{-\infty}^{a}}{\color{red}\displaystyle\int_{-\infty}^{b}} \! {\color{purple}f(x,y)} \, {\color{red}dy} \, {\color{blue}dx} \]
We can verify that the joint CDF starts at zero
\[{\color{purple}F(0,0)} = 0\]
and that the joint CDF collects all probabilities
\[\displaystyle\lim_{a \to \infty, b \to \infty} F(a,b) = 1\]
If need be, we can recover the joint PDF from the joint CDF as the mixed second-order partial derivatives
\[{\color{purple}f(x,y)} = \displaystyle\frac{{\color{purple}\partial^{2}}}{{\color{blue}\partial x} {\color{red}\partial y}} {\color{purple}F(x,y)}\]
What is the joint CDF for the In-n-Out setting?
Marginal Probabilities
The marginal cumulative distribution functions can be computed as
% marginal CDF \[\begin{array}{rcl} {\color{blue}F_{X}(a)} & = & \displaystyle\lim_{{\color{red}b \to \infty}} {\color{purple}F(a,b)} \\ {\color{red}F_{Y}(b)} & = & \displaystyle\lim_{{\color{blue}a \to \infty}} {\color{purple}F(a,b)} \\ \end{array}\]
Intuition: the marginal CDF is seeking to analyze the probabilities in just one variable regardless of the other variables, so ``eliminate’’ the other varibles by taking their limits to infinity.
We can verify that the marginal CDFs start at zero
\[{\color{blue}F_{X}(0)} = 0 \text{ and } {\color{red}F_{Y}(0)} = 0\]
and that the marginal CDFs collect all probabilities
\[\displaystyle\lim_{{\color{blue}a \to \infty}} {\color{blue}F_{X}(a)} = 1 \text{ and } \displaystyle\lim_{{\color{red}b \to \infty}} {\color{red}F_{Y}(b)} = 1\]
What are the marginal CDFs for the In-n-Out setting?
Marginal Probabilities
The marginal probability density functions can be computed as
\[\begin{array}{rcl} {\color{blue}f_{X}(x)} & = & {\color{red}\displaystyle\int_{-\infty}^{\infty}} \! {\color{purple}f(x,y)} \, {\color{red}dy} \\ {\color{red}f_{Y}(y)} & = & {\color{blue}\displaystyle\int_{-\infty}^{\infty}} \! {\color{purple}f(x,y)} \, {\color{blue}dx} \\ \end{array}\]Intuition: the marginal PDF is seeking to analyze the probabilities in just one variable regardless of the other variables, so ``integrate out’’ the other variables.
Alternatively,
\[{\color{blue}f_{X}(x) = \displaystyle\frac{d}{dx} F_{X}(x)} \text{ and } {\color{red}f_{Y}(y) = \displaystyle\frac{d}{dy} F_{Y}(y)} \]
What are the marginal PDFs for the In-n-Out setting?
Marginal Expectation
- What is the expected wait time to order food?
- What is the expected wait time to receive food?
Independence
Recall that two events \(A\) and \(B\) are independent if
\[{\color{purple}P(AB)} = {\color{blue}P(A)} \cdot {\color{red}P(B)}\]
Here, the variables \(X\) and \(Y\) in the In-n-Out example were independent, which can be easily verified by noting that the integrals were separable.
\[\begin{array}{rcl} {\color{purple}f(x,y)} & = & {\color{blue}f_{X}(x)} \cdot {\color{red}f_{Y}(y)} \\ {\color{red}\displaystyle\frac{1}{25}} {\color{blue}x}{\color{red}y}{\color{blue}e^{-x}}{\color{red}e^{-y/5}} & = & {\color{blue}xe^{-x}} \cdot {\color{red}\displaystyle\frac{y}{25}e^{-y/5}} \\ \end{array}\]and
\[\begin{array}{rcl} {\color{purple}F(a,b)} & = & {\color{blue}\displaystyle\int_{-\infty}^{a}} {\color{red}\displaystyle\int_{-\infty}^{b}} \! {\color{purple}f(x,y)} \, {\color{red}dy} \, {\color{blue}dx} \\ ~ & = & {\color{red}\displaystyle\frac{1}{25}} {\color{blue}\displaystyle\int_{0}^{a}} {\color{red}\displaystyle\int_{0}^{b}} \! {\color{blue}x}{\color{red}y}{\color{blue}e^{-x}}{\color{red}e^{-y/5}} \, {\color{red}dy} \, {\color{blue}dx}\\ ~ & = & {\color{red}\displaystyle\frac{1}{25}} {\color{blue}\left(\displaystyle\int_{0}^{a} xe^{-x} \, dx \right)} {\color{red}\left(\displaystyle\int_{0}^{b} \! ye^{-y/5} \, dy \right)} \\ \end{array}\]Upcoming lectures: dependent variables
Looking Ahead
due Fri., Mar. 10:
- WHW7
- LHW6
- Internet Connection (survey)
Exam 2 will be on Mon., Apr. 10
no lecture on Mar. 10, Mar. 24
