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Random Variable

A random variable is a real valued function whose domain becomes the complete sample space of a random experiment. 

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Random variables can be divided into two parts: Discrete random variable and Continuous  random variable.
A discrete random variable can take only a finite number of different values like 0, 1, 2, 3, 4, ..and so on, whereas a continuous random variable is a variable that can take an infinite number of possible values in a particular interval.

Concept of Random Variables includes the following important definitions and formulas:
1. The probability distribution of a random variable X is the system of numbers
X : x1, x2, ..., xn
P(X) : p1, p2, ... , pn
where, pi > 0,  and i varying from 1 to n, pi = 1, i = 1, 2, ..., n.
2. The probabilities pi, where P(X = xi) = pi, must satisfy the following:
1: 0 < pi < 1 for each i
2: p1 + p2 + ... + pk = 1.
3. Let X be a random variable with x1, x2, x3, ..., xn  as the possible values and p1, p2, p3, ..., pn  as the corresponding probabilities respectively. The mean of X, denoted by μ,
is the number ∑ xi pi, i = 1, 2,…,n. Mean of a random variable X is known as expectation of X, denoted by E (X).
4. Let X be a random variable having values x1, x2, ..., xn occur with probabilities p(x1), p(x2), ..., p(xn) respectively.
Let μ = E(X) be the mean of X. The variance of X, denoted by Var (X) or σx2, is defined as σx2 = Var (X) or equivalently σx2 = E (X – μ)2
5. The non-negative number σx = sqrt (Var (X)) is called the standard deviation of the random variable X.
6. The formula for variance: Var (X) = E (X2) – [E(X)]2
7. The Trials of a random experiment will be known as Bernoulli trials, if they  will satisfy the following points:
(i) The trials should be finite in numbers and should be independent.
(ii) Each trial can have 2 possibilities either success or failure.
(iii) In each trial, the probability of success remains equal.

Solved Examples:
Example 1: If we toss a coin ten times, then find the set of values and type of random variable?
Solution: Here, the random variable X will be the number of tails that can be noted down.
Therefore, X can only take the values 0, 1, ..., 10, so X is a discrete random variable.

Example 2: A bulb is burned until it is completely burnt out, then finds the values and type of random variable? 
Solution: Here, random variable X denotes the lifetime in hours and hence X could take any positive real value, so X is a continuous random variable.

Example 3: If we toss of a coin and define X = 1 if head comes up and X = 0 if tail comes up. Find the value that could be taken by random variable and its type.
Solution: Both event are equally likely: (X = 1) = (X = 0) = 1 /2  and hence, it’s a case of Bernoulli trials.

More topics in  Random Variable
Discrete Random Variable Continuous Random Variable
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