Probability Density Function

Used in the context of continuous Random Variable. See PMF for discrete random variables.

For a continuous [[random variable]] with [[probability density function|pdf]] $f$, we have
$$p(a \leq x \leq b)  = \int_a^bf(x)dx$$

In Probability, a PDF is a function whose value at any given point in the sample space can be interpreted as the relative likelihood that the value of the random variable would be equal to that sample.

These functions help model out the distribution of a particular variable, such as the age at which humans die, the height of a population, or suppose bacteria of a certain species typically live 4 to 6 hours. Then you can create a PDF.

for example, Normal Distribution has probability density $f(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$

PDF to CDF

• $F(x)={\int }_{-\infty }^{x}f(y)dy$

Example

x-axis is the height of an 20-yr old y-axis is the density.

if you integrate the PDF, you will always get 1.

Do not get these confused

Probability Density is NOT a probability. PMF gives you the probability, because you get discrete values.

With PDF, you can obtain the probability of a value on a particular range, by taking a slice of the function. You cannot the probability a a specific value, since the function is by nature continuous. See example below.

Example

Suppose bacteria of a certain species typically live 4 to 6 hours. The probability that a bacterium lives exactly 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.00… hours. However, you can quanitfy the probability that a bacteria dies between 5 and 5.01 hours.

For a PDF $f(x)$, we need to satisfy the condition $\int f(x)dx=1$ This is why it is possible for $f(x)$ to be $>1$, if on the x-axis the interval is $<1$, for example on interval $[0,\frac{1}{2}]$

Related

• See Distribution for conversions
• Probability Mass Function