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Statistics 101: Principles of Statistics11 chapters | 144 lessons | 9 flashcard sets

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Lesson Transcript

Instructor:
*Artem Cheprasov*

How can you find the expected value of something like height distributions? This lesson explains how to find and interpret the expected value of a continuous random variable.

Random processes are an inherent part of our world. Think about flipping a coin. Since the process is random by its very nature, there is no way to determine the outcome of a future coin flip. However, we can assign probabilities to future events of random processes. The mathematical construct that we utilize to achieve this goal is called a random variable.

In this lesson, you'll learn about the mathematical treatment of random processes and how to find and interpret the expected value of a continuous random variable.

**Random variables** designate the possible outcomes of random processes. They come in two types: discrete and continuous. A discrete random variable is associated with processes such as rolling a die and flipping a coin, in which there is a countable number of outcomes.

For example, we can define a random variable, *Z*, associated with rolling two 8-sided dice, as follows:

The random variable can take on the shown values because the lowest possible outcome is rolling a 1 on both dice, while the highest possible outcome is rolling an 8 on both dice.

On the other hand, a continuous random variable involves processes such as height and weight measurements, in which there is an infinite spectrum of possible outcomes. For example, the height of a person can be measured as 175.12 centimeters, but a more precise measurement may yield 175.117 centimeters, and an even more precise one may be 175.1168 centimeters, and so forth!

As an example, assume that the heights can range between 140 cm and 210 cm. Mathematically, the corresponding continuous random variable, *X*, would be written as an interval:

In other words, a continuous random variable does not have a countable number of possible outcomes.

The **probability density function**, which is another name for a continuous probability distribution function, is a graph of the probabilities associated with all the possible values a continuous random variable can take on. It appears as a continuous curve, with the random variable values plotted on the *x*-axis and their corresponding probabilities on the *y*-axis.

As shown in the Height Distribution graph, there is a continuous range of values between 140 cm and 200 cm.

A useful quantity that we can compute for a continuous probability distribution is the **expected value**. It is the outcome that we should expect to obtain on average.

The **expected value** of a continuous random variable can be computed by integrating the product of the probability density function with *x*. Mathematically, it is defined as follows:

We integrate over the interval in which *f(x)* is not equal to zero. As a simple example, assume that *f(x)* is defined as follows:

We can solve for the expected value, *E(x)*, as shown on the screen:

By plugging and chugging away, our answer becomes 18.7.

Referring back to the height distribution example, let's assume that *f(x)* is a Gaussian function with a mean, *mu*, of 170 and a standard deviation, *sigma*, of 10, as shown here:

We can set up the integral to compute the expected value as follows:

The actual computation of this integral is beyond the scope of this lesson, but this example should help you set up integrals for problems involving a Gaussian distribution.

Note that the expected value in itself may not have a high probability of occurrence.

However, the long-term average value of the probability distribution should be near the expected value.

To recap this lesson, we went over discrete and continuous random variables, as well as how to find and interpret the expected value associated with the probability density function.

**Random variables** are variables that designate the possible outcomes of random processes. Random variables can either be discrete, meaning that the number of possible outcomes is countable, or continuous, meaning that the number of possible outcomes is uncountable.

For example, a discrete random variable would describe processes such as flipping a coin or rolling dice. On the other hand, a continuous random variable would describe processes such as time measurements.

The probability distribution function for a continuous random variable, also called the **probability density function**, is a graph of the probabilities associated with all the possible values a continuous random variable can take on.

We have also covered how to calculate the **expected value**, which is the outcome that we should expect to obtain on average. For a continuous random variable, the calculation involves integrating *x* with the probability density function, *f(x)*. You should now feel comfortable solving problems similar to the examples we did.

**Random variable**: Random variables are discrete or continuous variables that designate the possible outcomes.

**Probability density function**: Probability density function is a graph of the probabilities associated with all the possible values a continuous random variable can take on.

**Expected value**: Expected value is the average outcome we could obtain.

If you fully understood the lesson above, you might easily:

- Name two types of random variables
- Find and interpret expected values of a continuous random variable

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Statistics 101: Principles of Statistics11 chapters | 144 lessons | 9 flashcard sets

- Go to Probability

- Graphing Probability Distributions Associated with Random Variables 6:33
- Finding & Interpreting the Expected Value of a Continuous Random Variable 5:29
- Probabilities as Areas of Geometric Regions: Definition & Examples 7:06
- Normal Distribution: Definition, Properties, Characteristics & Example 11:40
- Finding Z-Scores: Definition & Examples 6:30
- Estimating Areas Under the Normal Curve Using Z-Scores 5:54
- Estimating Population Percentages from Normal Distributions: The Empirical Rule & Examples 4:41
- Using the Normal Distribution: Practice Problems 10:32
- Using Normal Distribution to Approximate Binomial Probabilities 6:34
- How to Apply Continuous Probability Concepts to Problem Solving 5:05
- Go to Continuous Probability Distributions

- Go to Sampling

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