1. Let `X~N(mu, psi^2)` and `K` be a constant. Calculate

(a) `E(I{X>K})` where `I` is the indicator function.

(b) `E("max"{K-X,0})`.

Hint Use `(K-X) = (K-mu) - (X-mu)`.

(c) `E(e^{tx})`.

Leave answers where appropriate in terms of the normal cumulative distribution function
`Phi(·)`.


2. A random variable `Y` is said to be lognormally distributed with parameters `mu` and `sigma` if
`log Y ~ N(mu, sigma^2)`. We sometimes write `Y ∼"lognormal"(mu, sigma^2)`.

(a) Using your answer to Question 1(c), calculate `E(Y)` and `Var(Y)`.

(b) Suppose `mu = log f - 1/2 sigma^2` for a constant `f>0`.
Calculate `E(Y)` and show that `Var(Y) = f^2 sigma^2` (`1+{sigma^2}/2 + {sigma^4}/6 + `higher order terms).


3. A sequence of random variables `X_0, X_1, ..., X_n, ...` is defined by `X_0 = 1` and `X_n = X_{n-1}xi_{n-1}`, where `xi_i`, `i=0,1,...`, are independent and identically distributed with

\[
\xi_i =
\cases{
1+u & \text{with probability } p\cr
1+d & \text{with probability } 1 – p,
}
\]

where `u>d`.

(a) Calculate `E(X_n|X_{n-1})`.

(b) Let `Y_n = {X_n}/(1+r)^n` for a constant `r>0`.
Find in terms of `p`, `u`, and `d` the value of `r` such that `E(Y_n|Y_{n-1}) = Y_{n-1}`.
Hence find in terms of `u` and `d` the range of such possible values for `r`.

(c) Suppose `E(Y_n|Y_{n-1}) = Y_{n-1}` for all `n`.
Use any result concerning conditional expectation to prove that `E(Y_n|Y_m) = Y_m` for all `n>m>=0`.


4. Show that the first three non-zero terms of the Taylor series for the normal cumulative distribution function `Phi(x)` around zero are

`Phi(x) = 1/2 + x/sqrt{2pi} - {x^3}/{6sqrt{2pi}}`.

Thanks