MATH375 (Mock) Class Test
Time-length: 60 minutes. Worth 30% of the total mark.
1. (a) Give the definition of a probability measure. [4 marks]
(b) Let (Ω, F, P) be a probability space and X a random variable defined on it. If X(ω) = c for all ω ∈ Ω, i.e. X is a constant random variable, then prove that the σ-algebra generated by X is the trivial σ-algebra. [6 marks]
2. (a) Let (Ω, F) = ([0, 1], B[0, 1]). Define the probability measure P on this space as:
for some n > 0. Let the random variable X on this space be defined as X(ω) := 1 − ω for all ω ∈ Ω. Find the distribution measure µX[a, b], where 0 ≤ a < b ≤ 1. [5 marks]
(b) Let (W(t), t ≥ 0) be a standard Brownian motion. Compute:
[5 marks]
3. Let r, σ, T, S0, K, M, N, be given positive numbers, W(T) ∼ N(0, T), and consider the random variable:
Calculate the following expectation:
[10 marks]