Probabilities as Areas of Geometric Regions: Definition & Examples

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  • 0:01 Probability
  • 0:28 Defining Random…
  • 1:42 Probabilities as Areas…
  • 5:05 Expected Value of…
  • 6:11 Lesson Summary
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Lesson Transcript
Instructor: Artem Cheprasov

Artem has a doctor of veterinary medicine degree.

In this lesson, you're going to learn what a random variable is and examine core concepts related to probabilities as areas of geometric regions and expected values of probability distributions.


Have you ever tried to calculate your odds of winning a prize drawing? A lot of processes in the world are associated with randomness and uncertainty. In such cases, we cannot exactly determine the future outcome, but we can make predictions about how likely certain outcomes will be. In this lesson, you will learn about probability distributions as areas of geometric regions as well as how to find expected values.

Defining Random Variables & Processes

A random variable is the set of all possible outcomes associated with a random process. Depending on the process, a random variable may be discrete or continuous. A discrete random variable describes processes in which the total number of possible outcomes is countable, while a continuous random variable describes processes in which we cannot count the total number of possible outcomes.

For instance, we can use a discrete random variable to numerically describe any city's population in a given year. Although there are many possible outcomes, they are still countable. Each individual counts as one unit. It's not like a little kid counts as half a unit and someone who is 75 is 1.75 units. Each person is a distinct whole unit!

On the other hand, a continuous random variable could be utilized to describe the height distribution of a city's population. This is because height can be measured to an infinite precision, at least in theory. Someone can be said to be 70.2 inches tall, or 65.500000001, or 70.2384822 inches tall and so on.

Probabilities as Areas of Geometric Regions

The probability density function, f(x), must satisfy the following conditions:


When we ask about the probability of a particular outcome, we are referring to an area under the curve of the probability density function. That is, if x is the random variable associated with the probability density function, f(x), and a < b, then we have the formula shown on the screen:


So now, suppose we have the following probability density function:


The probability of having exactly outcome x_1, denoted P(X = x_1), is zero. This is because we are dealing with a continuous probability distribution, in which x_1 has an infinite precision and no width along the x-axis, as shown by the brown arrows in the figure. In order to estimate the probability of x_1, it's necessary to define a small interval near x_1, namely x_1 + dx. Now we have an area under the curve, with both height and width components that can be calculated using integration.


This, by the way, is the reason why f(x) is multiplied by dx in the following integral expression:


Let's look at an example. Suppose that the function f(x) describes the yearly income distribution within a community, in thousands of dollars, as follows:


Meaning, the probability that someone makes less than $18,000 per year is zero, while the probability for a yearly income greater than or equal to $18,000 varies as a function of x.

We shall now answer the following three questions:

1. What is the probability that a randomly selected person has a yearly income between $40,000 and $50,000? We calculate this probability by noting that 40 is less than or equal to x is less than or equal to 50, and subsequently plugging these limits into the integral just like you see now:


By plugging and chugging away, we get an answer of 0.015.

2. What is the probability that a randomly selected person has a yearly income of, at most, $30,000? In this case, x less than or equal to 30; we can solve this by breaking up the integral into a sum of two integrals:


Again by plugging and chugging away, we get an answer of 0.066.

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