In a random experiment there are more than one outcomes mostly. To each outcome, a probability is attached so that to find the likelihood of that outcome to occur. Probability distribution function is the listing of values of random variable to their respective probabilities. A probability distribution function defines the probability distribution. Probability distribution function can be of two types depending on the nature of sample space. If it is a discrete random variable, then the distribution function is called probability mass function else if it is continuous random variable then it is called a probability density function. 

Definition

A function which maps the members of a sample space to probabilities of their occurrence is known as probability mass function. The probability mass function is defined for discrete random variables, that is, random variable having discrete values. Suppose, a die is rolled. Rolling of die can have any one of six outcomes $\{1,\ 2,\ 3,\ 4,\ 5,\ 6\}$. The probability attached to each outcome can be defined using a function known as probability mass function. In this case, the probability mass function, $p.m.f$ = $\frac{1}{6}$.

Probability Density Function

For continuous random variables, the function mapping the values of the variable in a certain interval to probability of its occurrence is known as probability density function.For continuous random variables, the probability density function is the function which to be integrated on the given interval to get the probability of getting a value within that interval. The probability density function for getting the value of a random variable between $3$ and $13$ can be represented as $P(3\ <\ x\ <\ 13)$.

Probability Distribution Function

Probability distribution function is the function defining probability distribution. It is of two types: probability mass function and probability density function. The probability mass function is defined for discrete random variables and probability density function is defined for continuous random variables. The probability density function is a derivative of cumulative distribution function. The probability mass function does not need an integration but to get value of probability density function we do need integration. Just as in physics we need to integrate mass to get density, and that is why these terms mass and density are used for discrete and continuous random variables respectively.