# Developing Discrete Probability Distributions Empirically & Finding Expected Values

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Lesson Transcript
Instructor: Rudranath Beharrysingh
In this lesson, we will look at creating a discrete probability distribution given a set of discrete data. We will also look at determining the expected value of the distribution.

## Changing Data Into Probability

Imagine if you were given a set of numbers representing the number of cell phones in different households.

How would you interpret these numbers? In order to understand data, it is often necessary to tabulate the data. As data is collected, it also has to be decided whether the data is discrete or continuous since this will affect the way the data is tabulated. Let's take an example of a set of discrete data.

Suppose a research team gathers some data on the number of cell phones in households in the fictitious town of Phonyville. The first step would be to decide what the random variable represents. Since we are looking at the number of cell phones, it is natural to let X, the random variable, be the number of cell phones per household.

Also note that X is discrete because it represents distinct numbers, since one cannot really have a fraction of a cell phone! Thus, X could range from 0 to a very large number. However, let's assume the percentage of homes having 5 or more cell phones was negligible. If we tabulated the results, it could look like the following:

X = Number of Cell Phones Number of Households
0 457
1 18,763
2 22,345
3 12,534
4 9,830
5 or more 11
Total number surveyed: 63,940

If we look at the table, we see that there were 457 households with no cell phones, 18,763 households with 1 cell phone, 22,345 households with 2 cell phones and so on. Also, 5 or more is grouped as one category, since relative to the sample size, the amount of households with 5 or more cell phones was very small. Also notice that a grand total of 63,940 households were surveyed to collect this data.

So, we have a count of the number of households with the different amount of cell phones, but we need to convert these numbers to a percentage or probability. Well, we do this by dividing each count of the number of households by the total number of households to create a decimal or proportion for each category. The decimal is also an empirical probability.

X = Number of Cell Phones Number of Households P(X)
0 457 0.007147
1 18,763 0.293447
2 22,345 0.349468
3 12,534 0.196028
4 9,830 0.153738
5 or more 11 0.000172
Total surveyed: 63,940 Total of P(X): 1

Looking at the table, we see that the probability of a household with no cell phones (or probability that X = 0) is 457 divided by 63,940, which is 0.007147. The probability of selecting a household with only one cell phone (or probability that X = 1) is 18,763 divided by 63,940, which is 0.293447, or about 29%, and so on. If you want, you can pause the video to take a minute here and calculate the rest of the probabilities in the table to make sure your results match those in the table.

You should have calculated the same probabilities as the table and note the probability for 5 or more cell phones is very small at 0.00017, which is the reason 5 or more cell phones were grouped into one category. Also note that all the probabilities sum to a total of 1, or 100%, which is the property of a probability distribution.

So, we see empirical probabilities are probabilities generated from data and what we calculated was the probability distribution for X, the number of cell phones in households in the town of Phonyville!

## Expected Value of the Distribution

What if we randomly knocked at a home in Phonyville - how many cell phones would we expect the residents to have?

To calculate the expected value, we use the formula: expected value of X = Sum of (X times P(X))

For ease of calculation, I will round the above probabilities to three decimal places. Most of the time, this will not affect the results significantly, but when in doubt keep your probabilities with 5 to 6 decimal places in your calculations until the end. So, we expect (0 x .007) + (1 x .293) + (2 x .349) + (3 x .196) + (4 x .154) + (5 x .000) = 2.196 cell phones in a randomly selected household.

X = Number of Cell Phones Number of Households P(X) X times P(X)
0 457 0.007147 0.000
1 18,763 0.293447 0.293
2 22,345 0.349468 0.699
3 12,534 0.196028 0.588
4 9,830 0.153738 0.615
5 or more 11 0.000172 0.001
Expected Value = Sum of (X times P(X)) = 2.196

Thus, the expected value of the distribution is 2.196, or about 2.2. However, in reality you can't have a partial cell phone, and some households would have more and some less than 2.2, but this is the amount of cell phones per household that we would expect on average. In fact, another name for the expected value is the mean of the distribution.

Let's take another example.

## Insurance

How do insurance companies know what to charge you for a policy? How do they know that right after you obtain an insurance policy for a car, you are not going to crash and make a claim the very next day or in the near future? Or, if you take out that million-dollar life insurance policy, how do they (the insurance companies) know you won't die next week? It's a big risk for them to take! Or is it?

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