## 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 *a *discrete* *random variable*. *For 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 - If
*X*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,
*X*+*Y, X -Y, aX, bY*or a*X*+*by*are also random variables where*a*and b are any two constants except that*a*and b

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