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.
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The area under the curve of a density function over its entire range is equal to 1.
Density functions are non-negative, meaning $f(x) \geq 0$ for all $x$ in the domain.
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 $b$: $$P(a \leq X \leq b) = \int_a^b f(x) \, dx$$.
Common examples of density functions include the normal distribution and exponential distribution.
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.