Self Online Study  Mathematics  Probability  Probability DistributionRandom Variable: Let S be the sample space associated with a given random experiment A real – valued function X which assigns a unique real number X(w) to each w is belongs to S, is called random variable. A random variable which can assume only a finite number of value is called a discrete random variable. If a variable can take on any value between two specified values, it is called a continuous variable; otherwise, it is called a discrete variable. ]Some examples will clarify the difference between discrete and continuous variables. • Suppose the fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds. • Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable. Discrete Probability Distributions If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution. An example will make this clear. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the random variable X represent the number of Heads that result from this experiment. The random variable X can only take on the values 0, 1, or 2, so it is a discrete random variable. Continuous Probability Distributions If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution. A continuous probability distribution differs from a discrete probability distribution in several ways. • The probability that a continuous random variable will assume a particular value is zero. • As a result, a continuous probability distribution cannot be expressed in tabular form. • Instead, an equation or formula is used to describe a continuous probability distribution. Most often, the equation used to describe a continuous probability distribution is called a probability density function. Sometimes, it is referred to as a density function, a PDF, or a pdf. For a continuous probability distribution, the density function has the following properties: • Since the continuous random variable is defined over a continuous range of values (called the domain of the variable), the graph of the density function will also be continuous over that range. • The area bounded by the curve of the density function and the xaxis is equal to 1, when computed over the domain of the variable. • The probability that a random variable assumes a value between a and b is equal to the area under the density function bounded by a and b. For example, consider the probability density function shown in the graph below. Suppose we wanted to know the probability that the random variable X was less than or equal to a. The probability that X is less than or equal to a is equal to the area under the curve bounded by a and minus infinity  as indicated by the shaded area.
Mean And Variance of Random Variables: Let a random variable X assume values ${x}_{}$_{ 1} , x_{ 2} ,........., x_{ n} with probability ${p}_{}$_{ 1} , p_{ 2} ,......., p_{ n} respectively such that each ${p}_{}$_{ i} ≥0, Then mean of X denoted $\begin{array}{l}\mu \text{is defined as}\end{array}$ And the variance denoted by $\begin{array}{l}{\mathrm{\sigma 2\text{}\text{}is\; defined\; by}}^{}\\ {\sigma}^{\text{2}}=\left({\displaystyle \sum _{}^{}{\mathrm{xi}}_{}{\mathrm{pi}}_{}{\mu}^{2}}\right)\\ \text{Standard deviation}\sigma \text{is given by}\\ \\ \end{array}$

