Probability density function

Definition

A probability density function (PDF) describes the likelihood of a continuous random variable taking on a particular value. It is represented by a curve where the area under the curve within a given interval represents the probability that the variable falls within that interval.

5 Must Know Facts For Your Next Test

1. The total area under the curve of a PDF is always equal to 1, representing 100% probability.
2. A PDF can never take on negative values; it must be non-negative for all possible values of the random variable.
3. The probability that a continuous random variable exactly equals any specific value is zero; probabilities are only meaningful over intervals.
4. The integral of a PDF over an interval $[a, b]$ gives the probability that the variable falls within that interval: $P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx$.
5. Common examples of PDFs include the normal distribution, exponential distribution, and uniform distribution.

Review Questions

• What does the area under the curve of a probability density function represent?
• Can a PDF take on negative values? Explain your answer.
• How do you calculate the probability that a continuous random variable falls within an interval $[a, b]$ using its PDF?

Related terms

Cumulative Distribution Function (CDF): A function representing the probability that a random variable takes on a value less than or equal to a specific value. It is obtained by integrating the PDF from negative infinity to that value.

Normal Distribution: A type of continuous probability distribution for a real-valued random variable with symmetrical bell-shaped curve known as Gaussian distribution.

Exponential Distribution: A type of continuous probability distribution often used to model time until an event occurs, characterized by its decreasing exponential shape.