ne.gif (2791 bytes)     NE582 Monte Carlo

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Review for Test #1

For the first test, you should be able to: 
    1. Differentiate between wpe150.gif (1289 bytes), 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.
    3. Generate and find the period for a pseudo-random number stream from given values of a, b, m and wpe152.gif (870 bytes).
    4. Generate a few terms of a Halton sequence of any base.
    5. 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 continuous distribution using a Metropolis algorithm
    6. 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.
    7. Develop algorithms for choosing from any of the common PDFs from transport.

    The above will prepare you for 90% of the test.

    In addition, review your notes.  Up to 10% of the text will involve 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 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.

    Answer:
    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:

    Answer:
    1. Pull a random number squiggle
    2. x=SQRT(32*squiggle+4)

    Do NOT have an "instruction" like "Choose x from an exponential distribution between 0 and 4."


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