[Solution] A pair of fair dice are to be continually rolled until - #80168
1. Discrete random variables
A pair of fair dice are to be continually rolled until all possible outcomes of the sum of the two faces of the dice, i.e. 2, 3, 4, ...12 have occurred at least once. Develop a simulation study to estimate the expected number of dice rolls that are needed.
2. Monte Carlo simulation
Consider the following definite integral
\[I=\int\limits_{0}^{\pi /2}{\int\limits_{0}^{\pi /2}{\frac{\sin y\,dxdy}{\sqrt{1-{{\sin }^{2}}x{{\sin }^{2}}y}}}}\]
Provide Matlab code that does the following
• (10) Evaluates I using Monte Carlo simulation using it data points, where n = 256, 512 and 1024
• (10) calculates the mean squared error (MSE) for each of the values of it above, where the mean is obtained over 10 trials. Plot the MSE versus rt. Assume that I = \({{\pi }^{2}}/4\).
3. Poisson process
Buses arrive at a sporting event according to a Poisson process with rate 5 buses per hour. Each bus is equally likely to contain either 20,21, ... , 40 fans, with the numbers in the different buses being independent. Provide Matlab code that does the following
4. Continuous random variables
A casualty insurance company has 1000 policy holders, each of whom will independently present a claim in the next month with probability p = 0.05. Assume that the amounts of the claims made are independent exponential random variables with mean $800. Use simulation to estimate the probability that in any given month the sum of these claims exceeds $50,000.
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