Probability Theory is a beautiful subject. It might seem a bit obscure at first to some and if that's the case, don't despair and don't give up. You can also fire a few emails to me if you just can't get your head around it. Once you understand that, ultimately, probability is just a mathematical measure of uncertainty and once you start concentrating on understanding the phenomena rather than blindly applying formulas, you are on the path of becoming a successful thinking probabilist.

We may do only three supervisions, depending on progress and request.

Supervision 1:

• Problems 1-9 from Richard Gibbens' Problem Sheet.

Supervision 2:

• Problems 10-15 from Richard Gibbens' Problem Sheet.
• Show that for any discrete random variable that has a PGF, its probability mass function p(k) is given by p(k) = G^(k)(0) / k!, where G^(k)(z) is the k^th derivative of the PGF function G(z).
  
• Questions 1-3 from my Question list.

Supervision 3:

• Given a random variable X, either discrete or continuous, we define M(t) = E[e^tX], assuming this expectation exists. Show that the n^th moment of X, i.e. E[Xⁿ], in both discrete and continuous cases is given by E[Xⁿ] = M⁽ⁿ⁾(0), where M⁽ⁿ⁾(t) is the n^th derivative of M(t). For a discrete X, assume a probability mass function p(k). For a continuous X, assume a continuous probability density function f(x).
  
• Problems 5-9, 12, 13, 17, 19, 22 from my Question list.
  

Supervision 4:

• Questions 4, 10, 14, 16, 18, 21 from my Question list.
  NOTE: These are non-trivial, don't start the work late.
  

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