📚 Topic Summary
When dealing with multiple random variables, we often encounter joint probability mass functions (PMFs) for discrete variables and joint probability density functions (PDFs) for continuous variables. These functions describe the probability of observing specific combinations of values for these variables. Understanding their properties is crucial for calculating probabilities, expectations, and other statistical measures.
A joint PMF, denoted as $P_{X,Y}(x,y)$, gives the probability that the discrete random variable $X$ takes on the value $x$ and the discrete random variable $Y$ takes on the value $y$. A joint PDF, denoted as $f_{X,Y}(x,y)$, describes the relative likelihood that the continuous random variables $X$ and $Y$ take on specific values. Integration over a region in the $x$-$y$ plane gives the probability that $(X, Y)$ falls within that region.
🧮 Part A: Vocabulary
Match the term to its definition:
| Term |
Definition |
| 1. Marginal PMF |
a. A function describing the probability of a continuous random variable taking on a specific value. |
| 2. Joint PDF |
b. The integral of the joint PDF over all possible values of one variable. |
| 3. Independence |
c. A function describing the probability of discrete random variables taking on specific values. |
| 4. Marginal PDF |
d. The probability of one variable irrespective of the other. |
| 5. Joint PMF |
e. If the joint PDF or PMF can be factored into the product of individual PDFs or PMFs. |
(Answers: 1-d, 2-a, 3-e, 4-b, 5-c)
✏️ Part B: Fill in the Blanks
The sum of the joint PMF over all possible values must equal ______. The integral of the joint PDF over the entire space must equal ______. If two random variables are independent, then $f_{X,Y}(x,y)$ = ______ and $P_{X,Y}(x,y)$ = ______.
(Answers: 1, 1, $f_X(x)f_Y(y)$, $P_X(x)P_Y(y)$)
🤔 Part C: Critical Thinking
Explain, in your own words, how you would determine if two random variables, X and Y, are independent based on their joint PDF or PMF.