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MATH375 Class Test 1


1. Let (Ω , 于, P) be a probability space and let X and Y be two random variables defined on it.  If X(ω) = c for all ω ∈ Ω, i.e.  X is a constant random variable, then prove that X and Y are independent.


2. Let (Ω , 于) = ([0, 1], 夕[0, 1]). Define the set function P on this space as:

for some α > 0. Let the random variable X on this space be defined as X(ω) := 1 - ω for all ω ∈ Ω.


i) Find the value of α so that P is a probability measure.

ii) Find the distribution measure µx [a, b], where 0 s a s b s 1.


3. Let r, σ, T, S0 , K1 , K2 , be given positive numbers, W (T) ~ N (0, T), and con- sider the random variable:

Calculate the following expectation:


4. Let (W (t), t > 0) be a standard Brownian motion. Calculate: