Printable worksheet on Joint MGF properties and calculations

📚 Topic Summary

Joint Moment Generating Functions (MGFs) are powerful tools for analyzing the distributions of multiple random variables. They extend the concept of MGFs for single random variables, providing a way to compute moments and analyze the independence of random variables. Understanding their properties and how to perform calculations is crucial in probability and statistics. This worksheet will help you practice these skills!

🧠 Part A: Vocabulary

Match the following terms with their definitions:

1. Term: Joint MGF Definition: A function that uniquely determines the joint distribution of random variables.
2. Term: Independence Definition: When the joint MGF factors into the product of individual MGFs.
3. Term: Moment Definition: A quantitative measure of the shape of a probability distribution.
4. Term: Expectation Definition: The average value of a random variable.
5. Term: Random Variable Definition: A variable whose value is a numerical outcome of a random phenomenon.

Match each term with the correct definition.

✏️ Part B: Fill in the Blanks

Complete the following paragraph using the words provided below:

The Joint MGF, denoted as $M_{X,Y}(t_1, t_2)$, for random variables $X$ and $Y$ is defined as the ____________ of $e^{t_1X + t_2Y}$, where $t_1$ and $t_2$ are real numbers. If $X$ and $Y$ are ____________, then $M_{X,Y}(t_1, t_2) = M_X(t_1)M_Y(t_2)$. Finding the partial ____________ of the Joint MGF and evaluating at zero allows you to calculate the joint ____________ of X and Y.

Words: independent, expectation, derivatives, moments

🤔 Part C: Critical Thinking

Explain in your own words how you can use the joint moment generating function to prove that two random variables are independent. Provide a simple example to illustrate your explanation.