Statistics - Probability Density Function

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:

${P(a \le X \le b) = \int_a^b f(x) d_x}$

Where −

${[a,b]}$ = Interval in which x lies.
${P(a \le X \le b)}$ = probability that some value x lies within this interval.
${d_x}$ = 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) = \begin{cases} 1/24, & \text{for $ 0 \le x \le 240 $} \\ 0, & \text{otherwise} \end{cases} }$

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

Solution:

We have found the value of the following:

${P(14 \le X \le 14.45) = \int_{14}^{14.45} f(x) d_x \\[7pt] \ = \frac{1}{24} (14.45 - 14) \\[7pt] \ = \frac{1}{24}(0.45) \\[7pt] \ = 0.01875 }$