Random Variable and its types with properties

Random Variable

A random experiment is an experiment in which

  • All possible outcomes of the experiment are known in advance.
  • Any performance of the experiment results in an outcome that is not exactly known in advance.
  • The experiment can be repeated under identical condition.

More precisely we can say that,

A rule that assigns a real number to each outcome (sample point) of a random experiment is called a random variable (r.v.). It is governed by a function of the variable. Hence, a random variable is a real-valued function X (x) of the elements of the simple space (omega) where x is an element of (omega). Further, the range of the variable will be a set of real values. 

For example

In tossing a coin, x =1, if the coin falls with head, and x= 0 if the coin falls with a tail. The height of persons can be given by X (x)= X, the height is measured in centimetres or inches. A random variable is usually denoted by any of the capital Latin letters X, Y, Z, U, V etc.

Different types of random variables

There are two types of random variables namely 

  • Discrete random variable,
  • Continuous random variable.

Discrete random variable

A random variable, say x, which can take a finite number of values in an interval of the domain, is called discrete random variableFor example, if we toss a coin, the variable can take only two values 0 and 1 assigned to tail and head respectively, i.e.,

X(x)= 0(if x is tail) or 1(if x is head).

Continuous random variable

A random variable X, which can take any value in its domain or in an interval or the union of intervals on the real line, is called Continuous random variable. For a conti­nuous variable, the probability of a point is zero, i.e., P (X = x) =0.

Properties of a random variable

The properties of a random variable are,

  • If X is a random variable and a,b are any two constants, aX + b is also a random variable.
  • If is a random variable, X square is also a random variable.
  • If X is is a random variable, 1/X is also a random variable.
  • If X and Y are two random variables defined over the same space, Y, X -Y, aX, bY or aby are also random variables where and b are any two constants except that and b both are not zero.

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