What is a probability density function

A probability density function (PDF) describes the relative likelihood of a continuous random variable taking on values within specific ranges. Probabilities are found by integrating the function over an interval rather than at single points.

For a continuous random variable X, the PDF is denoted as f(x). The probability that X falls within an interval (a, b) is calculated by integrating the PDF over that interval.

$P(a<X<b)={\int }_a^{b}f(x)dx$

The probability at any exact single point equals zero, expressed as:

$P(X=c)=0$
$P(a<X<b)={\int }_a^{b}f(x)dx$

This occurs because the integral over a single point has no width, producing zero area under the curve.

What is the difference between a probability density function and a probability mass function?

A PDF applies to continuous variables, such as height, temperature, or time, where values can take any number within a range. A probability mass function (PMF) applies to discrete variables, such as coin flips, dice rolls, or counts, where values are distinct and countable.

The PDF requires integration over intervals to yield probabilities. The PMF assigns direct probabilities to individual outcomes, and these probabilities sum to one.

Feature
Probability Density Function
Probability Mass Function
Variable type
Continuous
Discrete
Probability calculation
Integration over intervals
Direct assignment to points
Probability at single point
Zero
Can be non-zero
Total probability
Integrates to 1
Sums to 1

What are the properties of a probability density function?

A valid probability density function must satisfy two fundamental properties: non-negativity and normalization.

Non-negativity

The function must be greater than or equal to zero for all values in its domain:

$f(x)\ge 0\text{for all }x$

This property prevents negative probabilities while allowing the curve to indicate relative likelihoods across different values.

Normalization

The total area under the curve must equal one:

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1$

This guarantees the probability of the variable falling anywhere in its entire range is 100%.

What is the relationship between a probability density function and a cumulative distribution function?

The PDF f(x) and cumulative distribution function (CDF) F(x) relate through differentiation and integration. The CDF accumulates probability up to a given value x as the integral of the PDF.

Obtaining the CDF from the PDF

Integrate the PDF from negative infinity to x:

$F(x)={\int }_{-\mathrm{\infty }}^{x}f(t)dt$

This integral yields the probability $P(X\le x)$.

Obtaining the PDF from the CDF

Differentiate the CDF to obtain the PDF:

$f(x)=\frac{d}{dx}F(x)$

This relationship holds where the derivative exists, following from the Fundamental Theorem of Calculus.

What are examples of probability density functions?

Common probability density functions include the normal, uniform, exponential, and gamma distributions.

Normal distribution

The normal distribution produces a bell-shaped curve symmetric around the mean μ with standard deviation σ:

$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$

This distribution models many natural phenomena, including human heights, measurement errors, and test scores.

Uniform distribution

The uniform distribution assigns equal density across an interval [a, b]:

$f(x)=\frac{1}{b-a}\text{for }a\le x\le b$

This distribution applies to random selections within fixed bounds, such as random number generation between two values.

Exponential distribution

The exponential distribution models time between events in Poisson processes:

$f(x)=\lambda e^{-\lambda x}\text{for }x\ge 0$

The parameter λ represents the rate of events. Applications include modeling customer arrivals, radioactive decay, and equipment failure times.

Gamma distribution

The gamma distribution generalizes the exponential distribution with shape parameter α and rate parameter β:

$f(x)=\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{x}^{\alpha -1}{e}^{-\beta x}\text{for }x>0$

The gamma function Γ(α) serves as a normalization constant. This distribution models waiting times for multiple events and rainfall amounts.