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Call the number of successes in

There are other probability distributions that a discrete random variable (I'll define this in more detail later) could take. We will discuss the

Also recall that if an experiment is replicated independently more and more times, the relative frequency of some occurrence approaches the probability that that occurrence will occur. Suppose that we are doing some experiment that is generating new data, independently. For instance:

```
1
```

1,4

1,4,2

1,4,2,4

1,4,2,4,3

1,4,2,4,3,1

1,4,2,4,3,1,1

and so forth. We could generate running averages as we go. Each new data point would be averaged in. The first average we have is just the average of the number 1, which is just 1. Then we have the average of 1 and 4, which is 2.5; the second in our sequence of averages is 2.5. The third one is the average of 1,4, and 2. This is 2.333 or so. The fourth is the average of 1, 4, 2, and 4, which is 11/4, and so forth.

We may also keep track of the relative frequency of each of the possible data values as we go. After the first data point has been received, we have a relative frequency of 1's which equals 1/1. After the second data point, the value 1 has a relative frequency of 1/2 and the value 4 has a relative frequency of 1/2. After the third, the value 1 has a relative frequency of 1/3, the value 2 has a relative frequency of 1/3, and the value 4 has a relative frequency of 1/3 also. And after the fourth, the value 1 has a relative frequency of 1/4, the value 2 has a relative frequency of 1/4, and the value 4 has a relative frequency of 2/4. After every new data point, we can compute the relative frequency of occurrences of all the data values.

Since we can think of each new data point (generated independently) as a new experiment, we know that as more and more experiments are done, the relative frequency should approach the probability of occurrence of that data value. But the average is the sum, over all the data values, of terms like

If we have a discrete random variable

When examining repeated independent random quantities, intuitively, it should be reasonable that if the relative frequencies approach their corresponding probabilities, that the sample mean should approach the expectation value (if it exists). In some sense, if you have enough data, it ought to be possible to be fairly sure that the sample mean is going to be close to the expectation value (if there is an expectation value). Results of this form are called "laws of large numbers"; their precise statement and proof are outside the range of our class. One such result is called the Weak Law of Large Numbers; another is called the (Kolmogorov) Strong Law of Large Numbers.

If you write a Bernoulli distribution as having the value 1 when a success occurs, and zero otherwise, you can calculate its expectation by

Here is another simple example involving sampling with replacement. Suppose we put 3 red marbles labeled "1" in a box, along with 2 white marbles labeled "2" and 5 blue marbles labeled "3". If I shake the box up and draw one at random, the number on the marble is a random variable. The probability of drawing a red "1" is 3/10, the probability of drawing a white "2" is 2/10, and the probability of drawing a blue "3" is 5/10. What is the expectation value of the number on the marble? You can calculate it by writing down all the values this random variable can take, multiplying it by the probability it will occur, and adding it all together. Try this as an exercize.

On to the Geometric distribution.

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