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Random Variable

A random variable X is a function whose domain is the sample space S and taking a value in the range set wh1ch is the real line with chance.

 

If the sample space consists of discrete elements then the r.v. is called discrete r.v. If the sample space consists of continuous elements then the r.v. is called continuous r.v. The distribution given by the random variable is called probability distribution. Again on the type of the r.v., the probability distribution is called discrete distribution or continuous distribution. Any discrete distribution is represented by probability mass function (pmj). For example,

IS a discrete distribution. Here the random variable X only takes the values -1, 0 and 1 with

probability 0.2, 0.4 and 0.4 respectively.

The characteristic of pmf is

(i) p(x) 2: 0 for all x

(ii) Σ p(x) = 1

 

Any continuous distribution is represented by probability density function (pd!J. For example,

                        f(x) = 1,                        O<x < l

= 0,                              elsewhere

 

• a continuous distribution. Here the random variable can take any value between 0 and 1 with probability = 1, for any other value the probability = 0.

The characteristic of pdf is

(i) f{x)>0                     for all x