Probabiltty Density Function

 Probability Density Function (Concept and Definition). 

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) dt represents the probability that X falls in the infinitesimal interval (x, x + dt). Symbolically

P (x≤X ≤ x + dt) = fx (x) dt 

Definition. 
p.d.f. fx (x) of the r.v. X is defined as:

\[fx(x)=\underset{\delta x\to 0}{\mathop{\lim }}\,\frac{p(x\le X\le x+\delta x)}{\delta x}\]

The probability for a variate value to lie in the interval dt is f(x) dt and hence the probability for a variate value to fall in the finite interval [α , β] is:

P(α≤X≤β)=∫_α^β f(x)dx

which represents the area between the curve y = f(x), x-axis and the ordinates at x = α . and x = β. Further since total probability is unity, we have ∫_a^b f(x)dx = 1, a where [a, b ] is the range of the random variable X . The range of the variable may be finite or infite. The probability density function (p.d.f.) of a random variable (r. v. ) X usually denoted by fx (x) or simply by f(x) has the following obvious properties

(i) f(x)≥0,    
(ii)∫_-∞^∞ f(x)dx=1

(iii) The probability P (E) given by
P(E)=∫_-E f(x)dx
is well defined for any event E.

Important Remark. 

In case of discrete random yariable the probability at a point, i.e., P (x = c) is not zero for some fixed c. However, in case of continuous random variables the probability at a point is always zero, Let P (x = c) = 0 for all possible values of c. This follows directly from  by taking α = β =c

This also agrees with our discussion earlier that P ( E ) = 0 does not imply that the event E is null or impossible event. This property of continuous r.v., viz.,

P(X= c)= 0, ∀ c

leads us to the following important result :
P(α≤X≤β) = P(α≤X<β) = P(α<X≤β) = P(α<X<β)i.e.,
in case of continuous r.v., it does matter whether we include the end points of the interval from α to β However, this result is in general not true for discrete random variables.