17.3. Independence of Random Variables

Try these first

1. 1.

   What is a random variable?

   Hint

   Think of what you measure in experiments.

2. 2.

   Can independent random variables affect each other's outcomes?

   Hint

   Reflect on the concept of independence.

3. 3.

   Two random variables are independent if:

   • A) They have the same marginal distribution
   • B) P(X = x
   • Y = y) = P(X = x)P(Y = y)
   • C) Cov(X
   • Y) = 0

   Hint

   Recall the definition of independence.

4. 4.

   If Cov(X, Y) = 0, then X and Y are definitely independent.

   Hint

   Consider examples of uncorrelated but dependent variables.

5. 5.

   Two people flip a coin simultaneously. Let A be the event of 'Person 1 gets heads' and B be 'Person 2 gets heads'. Are the events A and B independent? Justify your answer with probabilities.

   Hint

   Consider the outcomes of independent events.

6. 6.

   Given two dice rolls defined as random variables X and Y, show whether they are independent if the results are analyzed via conditional probabilities.

   Hint

   Think about how the outcome of each die is calculated.