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Information from a probability theory class for an m. Phil. In statistical science program. It includes three questions covering topics such as the probability of dice rolls, chebyshev's inequality, the borel-cantelli lemma, and martingales. Students are expected to find probabilities, prove mathematical inequalities, and apply the optional stopping theorem.
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Thursday 27 May 2004 1.30 to 3.
Attempt THREE questions. There are five questions in total.
The questions carry equal weight.
1 You throw 6n dice at random.
a) Find the probability that each number appears exactly n times.
b) Using Stirling’s formula, show that this probability is approximately Cn−^5 /^2 for some constant C to be found.
2 State and prove Chebyshev’s inequality.
State and prove the first Borel-Cantelli lemma.
Let X 1 , X 2 ,... , be independent identically distributed random variables with EX = a and VarX = σ^2 < ∞. Denote Sn =
∑n j=1 Xj^. Show that^ N^
− (^2) SN 2 → a (a.s.) as
N → ∞.
3 Suppose that S and T are independent exponential random variables with means 1 /α and 1/β respectively.
a) Find the distribution of min{S, T }.
b) Find the probability that S ≤ T.
c) Show that the two events {S ≤ T } and
min{S, T } ≥ t
are independent.
4 Let X 1 , X 2 ,... , be a sequence of independent identically distributed random variables with Pr(X = 1) = p = 1 − Pr(X = −1) and let Sn be the generated asymmetric simple random walk starting at 0, i.e., Sn =
∑n j=1 Xj^.^ Suppose^ p^ >^1 /2 and let
φ(x) =
1 −p p
)x .
a) Show that φ(Sn) is a martingale.
b) Denote Tx = inf
n : Sn = x
. State the Optional Stopping Theorem and use it to show that for a < 0 < b,
Pr
Ta < Tb
φ(b) − φ(0) φ(b) − φ(a)
c) For a < 0, show that Pr
minn Sn ≤ a
= Pr(Ta < ∞) =
1 −p p
)−a .
