 NE582 Monte Carlo

# Review for Test #1

For the first test, you should be able to:
1. Differentiate among the various statistical variables used: , including the formulas.  For a given distribution, be able to predict how many histories would be needed for a given standard deviation of the mean (or the standard deviation from a given number of histories).
2. Find the estimate of the  mean and the estimate of the standard deviation of the mean for a stream of estimates of a value, without using the statistical functions on your calculator.
3. Describe the meaning and importance of the Law of Large Numbers and the Central Limit Theorem.
4. Generate and find the period for a pseudo-random number stream from given values of a, b, m and .
5. Generate a few terms of a Halton sequence of any base.
6. Develop an algorithm for choosing x from a given pdf and domain using a technique I specify.   You should be prepared to handle a discrete distribution, a continuous distribution using a direct method, a continuous distribution using a rejection method, or a Metropolis algorithm.
7. Recognize a situation that calls for a probability mixing method for choosing x from a given distribution and develop an algorithm that implements the choice.

The above will prepare you for 90% of the test. Up to 10% of the test will involve ideas from the readings or creative application of the ideas we have discussed.

NOTE: When I say "an algorithm," I mean a series of basic steps that could be implemented by someone not knowing Monte Carlo.  It does NOT mean that you tell me how you figured it out.  Just figure it out and then tell me the result in an algorithmic form.

An algorithm usually begins with pulling one or more random numbers (which we can assume could be done) and ends with "Your chosen value of x is ____".

Example: Choose x from a flat distribution from 2 to 4.

1. Pull a random number squiggle.
2  Your chosen value is x=2+2*squiggle - 1.

Example: Choose x using a Direct Method from p(x)=x between 2 and 6

To work this problem out, you would have to:
Step 0. Normalize the distribution to find it must actually become p(x)=x/16.
Step 1. Find CDF to be P(x)=(x^2 - 4)/32  [Which passes the quick test that P(2)=0 and P(6)=1]
Step 2. Set equal to squiggle.
Step 3. Solve for x to get x=SQRT(32*squiggle+4)  [Which passes the quick test that squiggle =0 gives you 2 and squiggle =1 gives you 6]

All of this is your work.  None of it is the algorithm.  The algorithm is: