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# Discrete Random Variables

A discrete random variable is a variable which can only take a positive integer values and the sum of the probabilities is one. This type of random variables are obtained by counting and have for which there are no in-between values.

These values are typically the integers, for example number of students in a class may be 40 or 50 but not 40.6. In real world applications, the sample space for discrete
random variable has a finite number of possible values, and each has its own probability.

That is, it is possible to list all values in the sample space and to determine the probability of each.

A random variable is said to be discrete if it assumes only a finite and countably values.

Random variables are usually denoted by upper case letters and the possible values are by lower case letters like [Y = y].

The random variable Y is called discrete if it takes values in some countable subset {$y_1, y_2, y_3,....$} of R. The discrete random variable Y has mass function g : R [0, 1] given by g(y) = P(Y = y).

Probability Distribution of a Discrete Random Variable

If the possible values of a discrete random variables are y$_1$, y$_2$, y$_3$, …...., then P(y) is called a probability distribution of the discrete random variable ‘y’.

•     If P(yi) $\geq$ 0, for i = 1, 2, 3, …..
•    $\sum$ P(yi) = 1, otherwise

Mean of the distribution E(y) = $\sum$ y P(y).

Variance = E(y2) – (E(y))2 = $\sum$ y2 P(y) - $\sum$ (y P(y))2

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