Concept of random variables, discrete and continuous one and two dimensional random
 

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Concept of random variables, discrete and continuous one and two dimensional random

The outcome of an experiment need not be a number, for example, the outcome when a coin is tossed can be 'heads' or 'tails'. However, we often want to represent outcomes as numbers. A random variable is a function that associates a unique numerical value with every outcome of an experiment. The value of the random variable will vary from trial to trial as the experiment is repeated.
There are two types of random variable - discrete and continuous.
A random variable has either an associated probability distribution (discrete random variable) or probability density function (continuous random variable).
Examples

1. A coin is tossed ten times. The random variable X is the number of tails that are noted. X can only take the values 0, 1, ..., 10, so X is a discrete random variable.
2. A light bulb is burned until it burns out. The random variable Y is its lifetime in hours. Y can take any positive real value, so Y is a continuous 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.

Just like variables, probability distributions can be classified as discrete or continuous.
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.
The probability distribution for this statistical experiment appears below.
Number of heads Probability 0 0.25 1 0.50 2 0.25 The above table represents a discrete probability distribution because it relates each value of a discrete random variable with its probability of occurrence. In subsequent lessons, we will cover the following discrete probability distributions.

• Binomial probability distribution
• Hypergeometric probability distribution
• Multinomial probability distribution
• Poisson probability distribution

Note: With a discrete probability distribution, each possible value of the discrete random variable can be associated with a non-zero probability. Thus, a discrete probability distribution can always be presented in tabular form.
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.
Note: The shaded area in the graph represents the probability that the random variable X is less than or equal to a. This is a cumulative probability. However, the probability that X is exactly equal to a would be zero. A continuous random variable can take on an infinite number of values. The probability that it will equal a specific value (such as a) is always zero.