17. Partial Differential Equations

Try these first

1. 1.

   Define a random variable.

   Hint

   Think about the outcomes of a certain experiment.

2. 2.

   What does a Probability Mass Function (PMF) describe?

   Hint

   What type of variable is countable?

3. 3.

   What is the definition of independent random variables?

   • They influence each other's distribution
   • They do not affect each other's probabilities
   • They must have the same mean

   Hint

   Think about how one event might not influence another.

4. 4.

   For discrete variables, P(X=x, Y=y) = P(X=x) * P(Y=y) holds true if they are independent. True or False?

   • True
   • False

   Hint

   Recall the definition you studied.

5. 5.

   Given a joint distribution P(X,Y) = P(X)P(Y) where P(X=1) = 0.5, P(Y=1)=0.4, find P(X=1, Y=1).

   Hint

   Think about using the independence condition.

6. 6.

   Prove that two random variables are dependent if P(X=x,Y=y) does not equal P(X=x)P(Y=y).

   Hint

   Consider cases where the multiplication law fails.