In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.

Probability density function is defined by following formula:

P(a≤X≤b)=∫baf(x)dxP(a≤X≤b)=∫abf(x)dx

Where −

·        [a,b][a,b] = Interval in which x lies.

·        P(a≤X≤b)P(a≤X≤b) = probability that some value x lies within this interval.

·        dxdx = b-a

Example

Problem Statement:

During the day, a clock at random stops once at any time. If x be the time when it stops and the PDF for x is given by:

f(x)={1/24,0,for 0≤x≤240otherwisef(x)={1/24,for 0≤x≤2400,otherwise

Calculate the probability that clock stops between 2 pm and 2:45 pm.

Solution:

We have found the value of the following:

P(14≤X≤14.45)=∫14.4514f(x)dx =124(14.45−14) =124(0.45) =0.01875