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Discrete Probability Distributions: Equations & Examples

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  • 0:04 Selling Ice Cream Example
  • 1:08 Discrete Probability…
  • 2:17 Expected Value Function
  • 3:49 Variance & Standard Deviation
  • 6:00 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Discrete probability distributions give the probability of getting a certain value for a discrete random variable. In this lesson, you will learn how to calculate the expected value of a discrete variable and find the variance and standard deviation.

Selling Ice Cream Example

Let's say that your good friend James has just started a new business selling ice cream from an ice cream cart. His cart has a limited amount of space in it, so in the beginning, he decided to start each day with 100 servings of vanilla ice cream, 100 servings of chocolate ice cream, and 100 servings of strawberry ice cream. However, after a few weeks, he notices that on lots of days, he runs out of vanilla ice cream early and still has some strawberry and chocolate left. He has to go home and refill his ice cream cart with vanilla a long time before he runs out of the other flavors. He thinks he could make more money and eliminate his extra trips to resupply the ice cream cart if he could just figure out exactly how much of each type of ice cream to stock each day.

To solve his problem, James records the total amount of vanilla, chocolate, and strawberry ice cream he sold each day for two weeks. Each box of ice cream contains 10 servings, so James recorded the number of servings he sold in groups of 10 since he can't load just one serving at a time onto his cart. The data he collected is shown in the table below:


Ice cream data


Discrete Probability Distributions

James first wants to estimate how much vanilla ice cream he should put in his cart each morning, so he looks a little more closely at the data for vanilla ice cream. He records how many times each amount occurred during the last two weeks. Then, he calculates the probability that he will use a certain amount on any given day. For example, the probability that he will need 120 servings is 1/14, or 0.071.


Vanilla Ice cream probability data


The number of ice cream servings that James should put in his cart is an example of a discrete random variable because there are only certain values that are possible (120, 130, 140, etc.), so this represents a discrete probability distribution, since this gives the probability of getting any particular value of the discrete variable. If you add up all the probabilities, you should get exactly one. This is true for all discrete probability distributions.

0.071 + 0.071 + 0.143 + 0.143+ 0.214 + 0.071 + 0.143 + 0.143= 1.000

Expected Value Function

How can this information help James determine how many cases of vanilla ice cream to load in his cart each day? He can use the expected value function, which you can see below, in order to calculate how many ice cream cases he can expect to need on an average day. To use the expected value function, multiply each amount (xi) by the probability that he will use that amount of ice cream in a given day (pi). Then, add all of these value together.


expected value function


Using James's data, the expected value function gives the following:

E(x) = (120)(0.071) + (130)(0.071) + (140)(0.143) + (150)(0.214) + (160)(0.071) + (170)(0.143) + (180)(0.143) + (190)(0.143)

This simplifies down to the following:

E(x) = 8.57 + 9.29 + 20.00 + 34.29 + 21.43 + 12.14 + 25.71 + 27.14 = 158.57

The value given by the expected value function also represents the mean of the data set. This means that, on average, James can expect to need about 159 servings of vanilla ice cream in his cart each day. Since each box contains 10 servings, he can expect to use 16 boxes on an average day.

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