Simulation

• Example: Generate a distribution for the random variate:

What is the approximate probability that you will draw X ≤ 1.5?

Simulation

Simulation

Simulation

Simulation

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HHHHMMMM…

SimulationWhat if we crank up the

sample sizes?

Simulation

Simulation

Simulation

SimulationHere is how we did all of that:

Simulation

So what was the mean of that original Weibull distribution anyway?

Looks like that’s where the mean on the histograms is going

SimulationHow about the variance and sd?

= 0.376sWeibull= 0.613

= 1.5a = 1b

So what did we observe?

Looks like the distribution of the sample means is a little skewed, but they’re approaching normal

So what did we observe?

So what did we observe?

• Generalizing to a IID sample, it can be shown that:

The Central Limit Theorem

For a sum of IID random variables

For an average of IID random variables

Example

Random errors in the response of an instrument’s detector occur according to some unknown distribution, but have a mean = 2.45 units and variance = 0.63 units2.

Approximately, what is the probability that the cumulative error is greater that 123 units after 50 measurements?

Plot the approximate probability density for the cumulative error after 73 measurements.

Simulation

A a certain stock price moves with the approximate behavior from month to month:• Pr(no change) = 0.5• Pr(up 5%) = 0.25• Pr(down 5%) = 0.25

Plot a realization of the stock’s movement assuming an initial stock price of $10.

Approximately how much money to you expect to make/loose after one year of owning the stock? Three years? Ten years?