### PDF and CDF

• Understand relation between Probability Density Function (PDF) and Cumulative Distribution Function (CDF).

If X is continuous, the probability density function, f, is defined such that:

$( P( a\leq X \leq b) = \int_a^b f(x) dx$

The cumulative distribution function, F, of a discrete random variable X that has a probability distribution described by P(x$_i$) is defined as: $F(x_m) = \Sigma P(x_k) = P(X \leq x_m), m = 1,2,...,n$

If X is continuous, the cumulative distribution function, F, between a and b, is defined by:

$\int_a^b f(x) dx$

### Solved Example:

#### 9-1-01

Let X denotes the time a person waits for a bus to arrive as shown below. Calculate the probability that a person waits 90 seconds for the bus to arrive.

Solution:
90 seconds is 1.5 minutes. \begin{align*} P(X \leq 1.5) &= \mathrm{Area\ from\ } x=0 \mathrm{\ to\ } x=1.5\\ &= \mathrm{Area\ from\ 0\ to\ 1}\\ &+ \mathrm{Area\ from\ 1\ to\ 1.5}\\ &= 0.5 \times 1 + \dfrac{1}{2} \times 0.5 \times (0.5 + 0.375)\\ &= 0.719\ \mathrm{or\ } 71.9\% \end{align*}