Wiki Page, Math 13 - Perez Sandbox : Module 12 - Continuous Probability Distributions (2 of 2)

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2026-01-17T21:15:12-08:00

Previously, we examined the probability distribution for foot length. For foot length and for all other continuous quantitative variables, the probability distribution can be approximated by a smooth curve called a probability density curve.

These smooth curves are mathematical models associated with mathematical equations. The equations are beyond the scope of this course; therefore, we will only use the curve and not the equation to describe a probability distribution of a continuous quantitative variable.

The curve represents a probability distribution. The values on the horizontal axis are the variable values. The area under the curve represents probability.

As with any probability distribution, the probabilities sum to 1; therefore, the total area under the probability curve equals 1.

To find the probability that X is in an interval, find the area under the curve that is associated with the interval.

EXAMPLE

The probability curve below represents the probability distribution for foot length. X = foot length measured in inches.

If we want to find the probability that a randomly selected person has a foot length between 10 and 12 inches, we need to find the green area.

EXAMPLE

P(X < 9) denotes the probability that a randomly chosen male has a foot length of less than 9 inches. This probability is equal to the green area shown below.

EXAMPLE

What is the probability that the man's foot length is 9 inches? In other words, what is P(X=9)?

Answer: When the variable is a continuous quantitative variable, we cannot use the probability curve to determine the probability of a single outcome.

Why? Because there is no area over a single point.

On the other hand, if the variable was a discrete quantitative variable, such as shoe size, then we could use data and calculate a relative frequency to determine the probability that a person's shoe size is a 9. But with a continuous quantitative variable, P(X=9) is technically 0!

EXAMPLE

For a foot length, how does P(X<9) relate to P(X≤ 9)?

Answer: They are equal when we use the probability curve. The probability that a man has a foot length less than 9 inches is the same as the probability that he has a foot length less than or equal to 9 inches.

Why? Because visually, in terms of our probability curve, the area under the curve up to and including a certain point is the same as the area up to and excluding the point.

EXAMPLE

Empirical and theoretical probability distributions for female wrist girth

Wrist girth is a continuous quantitative variable. Here X is wrist girth measured to the nearest tenth of a centimeter.

The histogram below is an empirical probability distribution for female wrist measurements. The empirical probabilities are relative frequencies from the measurements of 206 women.

The red curve is the theoretical probability distribution based on a mathematical equation. The red curve approximates the histogram pretty well, with a similar shape, center and spread.

Now we will examine how to answer probability questions using the histogram and the curve.

What is the empirical probability that a woman's wrist measurement is less than 14 centimeters?

Answer: Add the probabilities in the two bins covering the interval X<14.
P(X<14) = 0.03 + 0.05 = 0.08. The empirical probability that a randomly selected woman has a wrist measurement less than 14 centimeters is 8%.

What is the theoretical probability that a woman's wrist measurement is less than 14cm?

Answer: Below we see only the curve without the histogram. We use the area under the curve associated with X<14. This is the area shaded in red, which is 12% of the total area. (We determined this using technology. We will show you how to do this later.)

Why is the theoretical probability greater than the empirical probability?

Answer: For X<14 you can see that the area under the curve is greater than the area in the first two bins. If we measure wrist girth to more decimal places, the empirical probability will tend to be closer to the theoretical probability.

Let's Summarize

We can use a probability density curve to represent the probability distribution of a continuous quantitative variable. These curves are mathematical models; they come from mathematical equations.

The probability that X has a value in a given interval is the area under the curve associated with the interval.

The total area under the probability curve is 1.

Comment

In case you are wondering ... probability density curves can have different shapes. The shape is comes from the mathematical equation we use. In choosing the curve, the goal is to approximate the shape of the empirical probability distribution that comes from data.

With all probability density curves, the area under the curve represents probability and the total area under the curve equals 1.

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