Discrete Random Variables - Cumulative & Probability Distribution

Discrete Random Variables

The numerical description for the outcome of an experiment is known as random variable. If a random experiment is carried out, there can be different outcomes each time the trial is done. When the outcomes comes as countable numbers then the random variable is known as discrete random variable. The probability distribution can be calculated for discrete random variable by obtaining the value of each probability for each value of discrete random variable.

Definition

A discrete random variable is the one which can take up discrete or countable values. For example, we toss two coins simultaneously and $X$ be a random variable representing the number of heads coming in the outcome. The number of heads can be $0,\ 1$ or $2$. Hence, the value of discrete random variable $X$ can be $\{0,\ 1,\ 2\}$.

Probability Distribution

Probability distribution of discrete random variables lists the probability for each value for the discrete random variable. There can be a graph, table or function representing the probability for each possible value of the random variable. If we are tossing two coins simultaneously with the random variable $X$, representing the number of heads coming as the outcome, the probability distribution can be given as the below table.

 $X$ $P(X)$ $0$ $\frac{1}{4}$ $1$ $\frac{1}{2}$ $2$ $\frac{1}{4}$

Cumulative Probability Distribution

Cumulative probability for a discrete random variable can be defined as the probability of getting a value less than or equal to a certain value of the random variable. IT can also be calculated as the probability of getting a value greater than or equal to a certain value. Similar to probability distribution of a discrete random variable, cumulative probability distribution can also be done.

Let us again take the example of tossing two coins simultaneously. If we have to calculate the cumulative probability of getting less than or equal to one head, we can calculate it as,

$P(X\ \leq\ 1)$ = $P(X = 0)\ +\ P(X = 1)$

Hence, cumulative probability = $\frac{1}{4}$ + $\frac{1}{2}$ = $\frac{3}{4}$

The cumulative probability distribution can in the tabular form as below:

 $X$ $P(X)$ $0$ $\frac{1}{2}$ $1$ $\frac{3}{4}$ $2$ $1$