MATH375: Tutorial 1
Tutorial 1
1. Let (Ω, F) = ([0, 1], B[0, 1]),and let
be two random variables.Let be a probability measure on (Ω, F) defined as
Find µX [a, b] and µY [a, b].
2. Let f (x) denote the standard normal density function, which is defined as:
Also let N(x) denote the standard normal cumulative distribution function, which is defined as:
Let (Ω, F, P) be a probability space on which a standard uniform. random variable Y is defined. Show that the random variable
is standard normal.
3. Let X be a random variable defined on (Ω, F, P) with exponential cumulative distribution function
where λ is a positive constant. Let be another positive constant, and define
Define as:
(i) Show that (Ω) = 1,
(ii) Derive {X ≤ x}, −∞ < x < ∞, i.e. the cumulative distribution function of X under .
(iii) Derive [X] and , i.e. the expected value and the variance of X under .