1. 511 Friday Feb. 23 2001 Math/Stat 511 R. Sharpley Lecture #15: Computer Simulations of Probabilistic Models

2. 511 Friday Feb. 23 2001 Probabilistic Simulation To perform probabilistic simulations using a Binomial random variable X, we first compute the cumulative distribution for X. In this example, we use 4 independent Bernoulli trials (n=4) with a probability of success p=1/2 on each trial.

3. 511 Friday Feb. 23 2001 Probabilistic Simulation To perform probabilistic simulations using a Binomial random variable X, we first compute the cumulative distribution for X. In this example, we use 4 independent Bernoulli trials (n=4) with a probability of success p=1/2 on each trial. We can think of this as stacking the probability mass function.

4. 511 Friday Feb. 23 2001 Probabilistic Simulation We then observe that the differences of the heights, which must add to one, partition the y-axis from 0 to 1 proportional to the probabilities.

5. 511 Friday Feb. 23 2001 Probabilistic Simulation We then observe that the differences of the heights, which must add to one, partition the y-axis from 0 to 1 proportional to the probabilities.

6. 511 Friday Feb. 23 2001 Probabilistic Simulation We then observe that the differences of the heights, which must add to one, partition the y-axis from 0 to 1 proportional to the probabilities. If we use a uniform distribution for [0,1] along the y-axis and take the inverse, we can then ‘pick’ values for X in correct proportion to their probabilities.

7. 511 Friday Feb. 23 2001 Probabilistic Simulation We then observe that the differences of the heights, which must add to one, partition the y-axis from 0 to 1 proportional to the probabilities. If we use a uniform distribution for [0,1] along the y-axis and take the inverse, we can then ‘pick’ values for X in correct proportion to their probabilities.

8. 511 Friday Feb. 23 2001 Probabilistic Simulation We then observe that the differences of the heights, which must add to one, partition the y-axis from 0 to 1 proportional to the probabilities. If we use a uniform distribution for [0,1] along the y-axis and take the inverse, we can then ‘pick’ values for X in correct proportion to their probabilities.

9. 511 Friday Feb. 23 2001 Probabilistic Simulation We then observe that the differences of the heights, which must add to one, partition the y-axis from 0 to 1 proportional to the probabilities. If we use a uniform distribution for [0,1] along the y-axis and take the inverse, we can then ‘pick’ values for X in correct proportion to their probabilities.

10. 511 Friday Feb. 23 2001 Probabilistic Simulation compute the cumulative distribution F X for X. This may involve: * *computing the probability mass distribution function f X (x) * *sum (or integrate) f to obtain the cumulative distribution F X (x) use a standard random number generator to produce random numbers between 0 and 1. use the inverse (F X ) -1 of these numbers to obtain values for x. Similar computer simulations can obtained for a general random variable X by performing the following steps: