Random Variables

Consider a sample space S of a random experiment. Then a random variable X(t) is a single real-valued function defined on the sample space S. For every number x, the probability is

where X(t) assumes x is defined.

Generally, we use a single letter X for the function X(t). The sample space S is the domain of the random variable X and the collection of all the numbers (values of X) is the range of the random variable X. Here, it may be noted that two or more different sample points might give the same value of X(t), but two or more different numbers in the range cannot be assigned to the same point.

If X is a random variable and x is a fixed real number, then we may define the event (X=x) as

For fixed real numbers x , x1 and x2, we can define the events in the following way:

The probabilities for the events in Equations 3.14 and 3.15 are denoted as

A random variable and its distributions are of two types such as discrete and continuous. A random variable X and its distribution is called discrete if X considers only finitely many or countable values x1 , x2 , x3 , ..., of X, with probabilities. p1 = P(X = x1), p2 = P(X = x2),

рз = p(x=x3) , ....

The distribution of X is obtained by the probability function f(x) of X as

From (3.17), we get the values of the distribution function F(x) as

This is a staircase or step function at the possible values xi of X are constant in between. Properties of p(x):

  • 1. 0 < p(xi)< 1, i = 1, 2,
  • 2. p(x) = 0 , if x Фx{ for i = 1, 2, ...;
  • 3. X/ (x ) = 1

Consider an experiment of tossing a fair coin thrice. The sample space S has a total of eight sample points, that is S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}, where H and T are heads and tails of the coin. If X is the random variable giving the heads obtained, then

The probabilities for these random variables are

A random variable X and its distribution are called continuous if in the distribution of X, F can be given by an integral of the following:

If F(x) is the distribution function, then

That means for every value of x and f(x) is continuous, the differentiation of F(x) gives f(x).


For an interval a < x < b, P(a < X < b) = F(b)- F(a)= I f (X)d^.


Properties of f(x):

  • 1. f(x) > 0;
  • ?
  • 2. I f (x )dx = 1;
  • —?
  • 3. f(x) is piecewise continuous;


4. P(a < X < b) = I f (x)dx.

a Mean and Variance of a Distribution

The mean or expectation of X, denoted by pX or E(X), is defined as

The variance o2 is defined as

The nth moment of the random variable X is


There are various distributions and these are of discrete or continuous type. Some of the standard distributions are as follows.

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