• A random variable is a variable that is subject to variations due to random chance. One can think of a random variable as the result of a random experiment, such as rolling a die, flipping a coin, picking a number from a given interval.

• The idea is that, each time you perform the experiment, you obtain a sample of the random variable. Since the variable is random, you expect to get different values as you obtain multiple samples.

• A probability distribution is a function that describes how likely you will obtain the different possible values of the random variable.

• It turns out that probability distributions have quite different forms depending on whether the random variable takes on discrete values (such as numbers from the set {1,2,3,4,5,6}) or takes on any value from a continuum (such as any real number in the interval [0,1]).

• Despite their different forms, one can do the same manipulations and calculations with either discrete or continuous random variables. The main difference is usually just whether one uses a sum or an integral.

Discrete probability distribution

• A discrete random variable is a random variable that can take on any value from a discrete set of values. The set of possible values could be finite, such as in the case of rolling a six-sided die, where the values lie in the set {1,2,3,4,5,6}

• However, the set of possible values could also be countably infinite, such as the set of integers {0,1,−1,2,−2,3,−3,…}

• The requirement for a discrete random variable is that we can enumerate all the values in the set of its possible values, as we will need to sum over all these possibilities.

• If we rolled two six-sided dice, and let X be the sum, then X could take on any value in the set {2,3,4,5,6,7,8,9,10,11,12}. The probability mass function for this X is plotted as a bar graph in the following figure.

Continuous probability distribution

• A continuous random variable is a random variable that can take on any value from a continuum, such as the set of all real numbers or an interval. We cannot form a sum over such a set of numbers. (There are too many, since such a continuum is uncountable.)

• Instead, we replace the sum used for discrete random variables with an integral over the set of possible values.

• For a continuous random variable X, we cannot form its probability distribution function by assigning a probability that X is exactly equal to each value. The probability distribution function we must use in the case is called a probability density function, which essentially assigns the probability that X is near each value. For intuition behind why we must use such a density rather than assigning individual probabilities, see the page that describes the idea behind the probability density function.