A random variable X is said to be continuous if it can take all possible alues between certain limits. In other words, a random variable is said to be continuous when its different values cannot be put in one to one correspondence with a set of positive integers.

A continuous random variable is a random variable that can be measured to any desired degree of accuracy. Examples of continuous random variables are age, height etc.

Consider the small interval (x,x+dx) of length dx round the point x. let f(x) be any continuous function of x so that f(x) dx represents the probability that x falls in the infinitesimal interval (x, x+dx).

Symbolically

P(x≤X≤x+dx) =fx(x)dx

The function fx(x) so defined is known as probability density function or simply density function of random variables X and is usually abbreviated as p.d.f. the expression, f(x) dx, usually written as dF(x), is known as the probability differential and the curve y = f(x) is known as the probability density curve or simply probability curve.

The properties of the probability density function (p.d.f) of a random variable X usually denoted by fx(x) or simply by f(x) is

- (i) f(x) ≥ 0, -∞ < x < ∞
- (ii)
- (iii) The probability P(E) given by

is well defined for any event E.

The formula for these measures in case of discrete frequency distribution can be easily extended to the case of continuous probability distribution by simply replacing pi = fi/N by f(x) dx, xi by x and the summation over ‘i’ by integration over the specified range of the variable X.

Let fx(x) or f (x) be the p.d.f of a random variable X where X is defined from a to b. then

is called the distribution function (d.f) or sometimes, the cumulative distribution function(c.d.f.) of the random variable X.

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