Density function

Definition

A density function is a function that describes the probability distribution of a continuous random variable. It provides the relative likelihood that the value of the variable lies within a particular range.

5 Must Know Facts For Your Next Test

1. The area under the curve of a density function over its entire range is equal to 1.
2. Density functions are non-negative, meaning $f(x) \geq 0$ for all $x$ in the domain.
3. The probability that a continuous random variable falls within an interval $[a, b]$ is given by the integral of the density function from $a$ to $b$: $$P(a \leq X \leq b) = \int_a^b f(x) \, dx$$.
4. Common examples of density functions include the normal distribution and exponential distribution.
5. The mean (expected value) of a continuous random variable can be found using $$E(X) = \int_{-\infty}^{\infty} x f(x) \, dx$$.

Review Questions

• What is the total area under the curve of any density function?
• How do you calculate the probability that a continuous random variable falls between two values using its density function?
• What is one common example of a density function?

Related terms

Probability Density Function (PDF): A specific type of density function used in statistics to describe the likelihood of different outcomes for a continuous random variable.

Cumulative Distribution Function (CDF): A function representing the probability that a random variable takes on a value less than or equal to a given point.

Expected Value: \(E(X)\), which represents the mean or average outcome expected if an experiment is repeated many times.