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**Discrete Random Variables and their Probability Distribution**

A discrete random variable is a variable whose value is determined by the outcome of a random experiment. A discrete random variable assumes values that can be counted. A continuous random variable is not countable. These variables can assume values only within a certain range or interval.

The probability distribution of a discrete random variable "$ \left( x \right) $" lists all the values that the random variable can assume and their corresponding probabilities. For example:

"$ \begin{align} Z = \dfrac {X - \mu}{\sigma} &&& t = \dfrac {\bar x - \mu_0}{\frac {s}{\sqrt {n}}}, \end{align}$"

The probability distribution of a discrete random variable is the third column above. It is derived by adding the frequency and dividing the individual frequencies by the total frequency.

Probability distribution has two characteristics or conditions:

- "$ x $" falls between "$0$" and "$1$" or "$0 \leq P \left( x \right) \leq 1$" and;
- "$ \sum p\left( x \right) = 1$".

From the above, we can see that the probability that a family owns **2** vehicles is **0.425** and the probability that a family owns **4** vehicles is **.080**. The probability that a family owns more than two vehicles is **0.245** plus 0.080 = 0.325.

The mean of a probability distribution is also called the expected value and it is the value that we expect to obtain if we repeat the experiment a large number of times. The mean of x in the above example is:

"$ \begin{align} E \left( x \right) &= \left( 0 \right) \ast .015 + \left( 1 \right) \ast \left( .235 \right) + \left( 2 \right) \ast \left( .425 \right) + \left( 3 \right) \ast \left( .245 \right) + \left( 4 \right) \ast \left( .080 \right) \\ &= 0 + .235 + .85 + .735 + .32 \\ & = 2.14 \end{align} $"

The mean for the probability distribution can also be stated as: 2.14.

**Poisson Probability Distribution**

This is another important probability distribution of a discrete random variable that has a large number of applications. Suppose a tractor breaks down on average about three times per month. We may want to find the probability of exactly two breakdowns in the next month. Each breakdown is called an occurrence in Poisson probability distribution terminology. Poisson probability distribution is applied to experiments with random and independent occurrences. The occurrences are random in the sense that they do not follow any particular pattern and therefore are unpredictable. Independence of the occurrences means that one occurrence does not influence the success occurrences or non-occurrences.

**Binomial Distribution**

In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. It is used and applied to find the probability that an outcome will occur x times in n performances. The binomial distribution is the basis for the popular binomial test of statistical significance. For example, given that the probability is .05 that a tool manufactured will be defective, we are interested in finding the probability that in a random sample of five of such tools manufactured, exactly one will be defective. The binomial distribution can be used in these cases.

Next - Continuous Probability Distribution – Normal Distribution