代写MATH375 (Mock) Class Test调试R语言程序

MATH375 (Mock) Class Test

Time-length: 60 minutes. Worth 30% of the total mark.

1. Let (Ω, F, P) be a probability space on which the random variable X is defined. Also let E[|X|] < ∞, and let G be a sub-σ-algebra of F.

(i) Give the definition of conditional expectation of X given . [4 marks]

(ii) Show that E[X] and E[X|] are independent. [6 marks]

2. Let X be a random variable defined on (Ω, F, P) with exponential cumu-lative distribution function

where λ is a positive constant. Let be another positive constant, and define

for which it can be shown that E[Z] = 1. The probability measure is defined as:

Let Y (ω) := θX(ω), where θ is a positive constant. Derive:

i) {Y ≤ y}, where −∞ < y < ∞, [7 marks]

ii) [Y ] and  [3 marks]

3. Let r, σ, T, S0, K1, K2, be given positive numbers, and (W(t), t ≥ 0) a standard Brownian motion. Consider the random variable:

Calculate the following expectation:

[10 marks]