What makes a density function

Now, you could imagine randomly selecting, let's say, hamburgers advertised to weigh a quarter-pound. If you weighed the hamburgers, and created a density histogram of the resulting weights, perhaps the histogram might look something like this:.

In this case, the histogram illustrates that most of the sampled hamburgers do indeed weigh close to 0. Now, what if we decreased the length of the class interval on that density histogram? Then, the density histogram would look something like this:. Now, what if we pushed this further and decreased the intervals even more? Now, you might recall that a density histogram is defined so that the area of each rectangle equals the relative frequency of the corresponding class, and the area of the entire histogram equals 1.

In the case of this example, the probability that a randomly selected hamburger weighs between 0. Now that we've motivated the idea behind a probability density function for a continuous random variable, let's now go and formally define it. The probability density function " p. As you can see, the definition for the p. Let's test this definition out on an example. To change or withdraw your consent choices for Investopedia. At any time, you can update your settings through the "EU Privacy" link at the bottom of any page.

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Key Takeaways Probability Density Functions are a statistical measure used to gauge the likely outcome of a discrete value e. PDFs are plotted on a graph typically resembling a bell curve, with the probability of the outcomes lying below the curve. A discrete variable can be measured exactly, while a continuous variable can have infinite values. Compare Accounts. The offers that appear in this table are from partnerships from which Investopedia receives compensation. This compensation may impact how and where listings appear.

Investopedia does not include all offers available in the marketplace. Related Terms Uniform Distribution Uniform distribution is a type of probability distribution in which all outcomes are equally likely.

Learn how to calculate uniform distribution. Note that as the width of the interval decreases, the area, and thus the probability of the length falling in the interval decreases. This also implies that the probability of the length of one randomly selected fish having a length exactly equal to a specific value is zero. This is because the area of a line is zero. Example: A probability density function is defined as. For f x to be a valid probability density function, what is the value of a?

Solution: To be a valid probability density function, all values of f x must be positive, and the area beneath f x must equal one. The first condition is met by restricting a and x to positive numbers. To meet the second condition, the integral of f x from one to ten must equal 1.