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Relative Frequency
Suppose that X is a random variable for the experiment, taking values in a space S. Note that X might be the outcome variable for the entire experiment, in which case S would be the sample space. Recall that the distribution of X is the probability measure on S given by

P(A) = P(X A) for A S.

Suppose now that we fix A. Recall that the indicator variable IA takes the value 1 if X is in A and 0 otherwise. This indicator variable has the Bernoulli distribution with parameter P(A) above.

1. Show that the mean and variance of IA are given by

E(IA) = P(A).
var(IA) = P(A)[1 − P(A)].
Now suppose that we repeat the basic experiment indefinitely to form independent random variables X1, X2, ..., each with the distribution of X. Thus, for each n, (X1, X2, ..., Xn) is a random sample of size n from the distribution of X. The relative frequency of A for this sample is

Pn(A) = #{i ****1, 2, ..., n****: Xi A**** / n for A S.

The relative frequency of A is a statistic that gives the proportion of times that A occurred, in the first n runs.

2. Show Pn(A) is the sample mean from a random sample of size n from the distribution of IA. Thus, conclude that

E[Pn(A)] = P(A).
var[Pn(A)] = P(A)[1 − P(A)] / n
Pn(A) �¨ P(A) as n �¨ �‡ (with probability 1).
This special case of the strong law of large numbers is basic to the very concept of probability.

3. Show that for a fixed sample, Pn satisfies the axioms of a probability measure.

The probability measure Pn gives the empirical distribution of X, based on the random sample. It is a discrete distribution, concentrated at the distinct values of X1, X2, ..., Xn. Indeed, it places probability mass 1/n at Xi for each i, so that if the sample values are distinct, the empirical distribution is uniform on these sample values.

Several applets in this project are simulations of random experiments with events of interest. When you run the experiment, you are performing independent replications of the experiment. In most cases, the applet displays the relative frequency of the event and its complement, both graphically in blue, and numerically in a table. When you run the experiment, the relative frequencies are shown graphically in red and also numerically.

4. In the simulation of Buffon's coin experiment, the event of interest is that the coin crosses a crack. Run the experiment 1000 times with an update frequency of 10. Note the apparent convergence of the relative frequency of the event to the true probability.

5. In the simulation of Bertrand's experiment, the event of interest is that that a "random chord" on a circle will be longer than the length of a side of the inscribed equilateral triangle. Run the experiment 1000 times with an update frequency of 10. Note the apparent convergence of the relative frequency of the event to the true probability.

Hi Chrome maybe this will help.
Cheers
baco