Self Online Study - Mathematics - Probability - Probability Distribution

Random 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 x-axis 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
$\begin{array}{l}{\mathrm{^{\sigma 2\; <mtext></mtext><mtext></mtext>}is\; defined\; by}}^{}\\ {\sigma }^{\text{2}}=\left({\displaystyle \sum _{}^{}{\mathrm{x_i}}_{}{\mathrm{p_i}}_{}-{\mu }^{\mathrm{^2\;}}}\right)\\ \text{Standard deviation}\sigma \text{is given by}\\ \mathrm{<span><img\; title="SOSNET\_CON/ReadingImages/985X3.jpg"\; src="SOSNET\_CON/ReadingImages/985X3.jpg"></span>}\end{array}$