# What is the difference between discrete probability distribution and continuous probability distribution?

Dec 20, 2017

See below.

#### Explanation:

A random variable is a real-valued function defined over a sample space. Consequently, a random variable can be used to identify numerical events of interest in an experiment. Random variables may be either continuous or discrete.

A random variable $Y$ is said to be discrete if it can assume only a finite or countably infinite* number of distinct values.

A set of elements is said to be countably infinite* if the elements in the set can be put into one-to-one correspondence with the positive integers.

Because certain types of random variables occur so frequently in practice, it is useful to have at hand the probability of each value of ra random variable. This collection of probabilities is called the probability distribution of the discrete random variable.

• Distribution functions for discrete random variables are always step functions

Example: Binomial distribution function, $n = 2 , \text{ } p = 1 / 2$

On the other hand, a random variable $Y$ is said to be continuous if it can take on any value in an interval. More precisely, a random variable $Y$ with distribution function $F \left(y\right)$ is said to be continuous if $F \left(y\right)$ is continuous for $- \infty < y < \infty$.

Unfortunately, the probability distribution for a continuous random variable cannot be specified in the same way as outlined above for a discrete random variable; it is mathematically impossible to assign nonzero probabilities to all points on a line interval while satisfying the requirement that the probabilities of the distinct possible values sum to one.

Rather, we define a probability density function for the random variable:

Let $F \left(y\right)$ be the distribution function for a continuous random variable $Y$. Then $f \left(y\right)$, given by

$f \left(y\right) = \frac{\mathrm{dF} \left(y\right)}{\mathrm{dy}} = F ' \left(y\right)$

wherever the derivative exists, is called the probability density function for the random variable $Y$.

Example: A distribution function $F \left(y\right)$ for a continuous random variable