1. Let X be a random variable with µ = E(X) and σ2 = V (X). Define X∗ = (X -µ)/σ. The random variable X∗ is called the standardized random variable associated with X. Show that this standardized random variable has expected value 0 and variance 1.

2. Peter and Paul play Heads or Tails. Let Wn be Peter's winnings after n matches. Show that E(Wn) = 0 and V (Wn) = n.

3. Find the expected value and the variance for the number of boys and the number of girls in a royal family that has children until there is a boy or until there are three children, whichever comes first.