CFRM 505: Monte Carlo Methods in Finance (Winter 2024)
HW 1
1. (Probability Essentials) Suppose the continuous random vector (X, Y ) has the joint probability distribution
for some constant c.
Complete the following parts analytically (i.e. not by simulation):
(a) Calculate P(X + Y > 1.5).
(b) Calculate E[Y ].
(c) Calculate Cov(X, Y ). Recall: Cov(X, Y ) = E[XY ] − E[X]E[Y ].
(d) What is f(x|y), the conditional probability density function of x given y?
(e) Compute E[X|Y = 1/3].
(f) Calculate P(X < 0.5|Y > 0.5).
Remark: Conditional probability and conditional expectation are very important concepts that will be used repeatedly in various stochastic settings (discrete/continuous random variables, discrete-time/continuous-time stochastic processes) throughout this course.
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For all simulation questions, append the full (Python or R) codes to your answers in your submission. The codes must be able to directly generate the answers. Present your codes and answers clearly for full credits.
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2. Consider the integral 
xy dydx. First, write the integral as an expectation along with the associated random variables. Then, implement MC simulation to estimate it. Try to use a sample size no less than 1000 for each variable. No need to compute by hand.
3. Consider the covariance Cov(U, 1 − U), where U is uniform on (1, 2). First, write the covariance in terms of expectations along with the associated random variables. Then, implement MC simulation to estimate the expectations to obtain the covariance. Use a sample size no less than 10,000. No need to compute by hand.
4. Consider the random variable defined by X = min(U1, U2), where U1 and U2 are IID U(0, 1). Implement MC simulation to estimate (i) the mean of X and (ii) the probability P(X > 0.5). Use a sample size no less than 10,000 for each problem. No need to compute by hand.