15. Partial Differential Equations

Try these first

1. 1.

   What is the joint probability density function (pdf)?

   Hint

   Think about how two random events relate.

2. 2.

   Define marginal distribution.

   Hint

   What do we focus on when we 'marginalize'?

3. 3.

   What is a marginal distribution?

   • A function for two variables
   • Distribution of one variable
   • Sum of joint probabilities

   Hint

   Think about the process of simplifying.

4. 4.

   True or False: Marginal distributions can reconstruct joint distributions.

   • True
   • False

   Hint

   Consider independence in your reasoning.

5. 5.

   Given the joint pdf 𝑓(𝑥,𝑦) = 12𝑥𝑦 for 0 < 𝑥 < 1 and 0 < 𝑦 < 1, calculate the marginal distributions of 𝑋 and 𝑌.

   Hint

   Focus on integrating each variable's contribution.

6. 6.

   Discuss the independence condition for two random variables and provide an example illustrating this with marginal pdfs.

   Hint

   Consider if separating their probabilities affects their joint behavior.