# Probability Density Function: Definition and Examples

Probability Density Function (PDF) provides the likelihood that the value of a random variable will fall between a certain range. PDF typically is used for continuous random variables. For discrete random variables, we use probability mass function.

## Definition of Probability Density Function (PDF)

Probability Density Function (PDF) is defined as the probability of a value of the random variable in the span of Δx. Note that, Δx should be very small (close to zero).

$f_X(x)= \lim\limits_{\Delta x \rightarrow 0} \frac{Pr[x≤X≤x+\Delta x]}{\Delta x}$

We do not say PDF returns the probability at point x is because the probability at any specific point for a continuous distribution will always be zero as the area under a point is 0.

That is, we connect PDF and probability via a very short range, namely Δx →0.

## Example 1: PDF for Normal Distribution

The following is the PDF of normal distribution.

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

If we use mean μ = 0 and standard deviation σ = 1, we can get the following PDF.

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

We can use R to plot this PDF. The following is the R code.

# generate a sequence of 50 numbers in the range of -3 and 3
x <- seq(-3, 3, length=50)

# use the normal distribution function dnorm() from R
y <- dnorm(x)

# plot x and y with a connected line
plot(x,y, type = "l")

The following is the plot of the PDF of normal distribution in R.

## Example 2: PDF for Uniform Distribution

The following is PDF for uniform distribution.

$f(x) = \left\{ \begin{array}{rcl} \frac{1}{b-a} & \mbox{for} & a \le x \le b\\ 0 & \mbox{for} & x < a \ or \ x>b \end{array}\right.$

If we use a=-2, and b=2, we can get the following PDF.

$f(x) = \left\{ \begin{array}{rcl} \frac{1}{4} & \mbox{for} & -2 \le x \le 2 \\ 0 & \mbox{for} & x < -2 \ or \ x>2 \end{array}\right.$

Similarly, we can use R to plot the uniform distribution PDF. Below is the R code.

# generate a range of data for x-axis
x <- seq(-4, 4, length=1200)

# calculate density for uniform distribution
y <- dunif(x, min = -2, max = 2)

# plot the uniform distribution
plot(x, y, type = 'l',lwd = 2)

The following is the plot of the PDF of uniform distribution in R.