Computer Lab supervisions - Probability

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], assuming this expectation exists. Show that the^{tX}*n*^{th}moment of*X*, i.e. E[*X*], in both discrete and continuous cases is given by E[^{n}*X*] =^{n}*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.