9.3. Practice Problems

Try these first

1. 1.

   What is the expectation of rolling a fair 4-sided die?

   Hint

   Use the formula for discrete expectations.

2. 2.

   Find the expectation of the number of heads in two tosses of a fair coin.

   Hint

   List down the possible outcomes.

3. 3.

   What does the Expectation of a random variable represent?

   • A frequency
   • An average value
   • A probability

   Hint

   Consider what we mean by the 'long-run average'.

4. 4.

   True or False: The expectation of a constant is always equal to the constant itself.

   • True
   • False

   Hint

   Review the properties of expectation.

5. 5.

   Let X be the number of defective items in a sample of size n. Given a fixed probability p, formulate E(X) in a binomial context.

   Hint

   Refer back to the binomial expectations formula!

6. 6.

   In a stochastic system, develop a PDE using E[u(x, t)] and show how it simplifies to a deterministic equation.

   Hint

   Think about how taking averages can smooth out variability.