### 4. Probability Distributions: Probability Models

### Continuous Probability Models

Continuous Probability Model

If the sample space #\Omega# of a random experiment consists of an *uncountable *set of outcomes, we must use a **continuous probability model **to assign probabilities to events.

Intervals are examples of **uncountable **sets. For example the interval #(0, \infty)# or the interval #[1,5]#.

Density Curve

The graph of a continuous probability model is called a **density curve**, and has the following properties:

- A density curve is always on or above the horizontal axis.
- The total area under the whole density curve always equals #1#.

Continuous Model: Calculating the Probability of an Event

For a continuous probability model, the probability of an event is the *area under the density curve* above all outcomes in that event.

Let #X# denote a number randomly generated from the interval #[0, 5]#.

Calculate the probabilities of the following events:

- #A = \{"\text{number}\gt 3"\}#
- #B = \{"\text{number between } 1 \text{ and }4"\}#

Since all numbers in the interval #[0,5]# have an equal chance of selection, the density curve is flat:

It is a rectangle with a width of #5#. This means that its height must be #\frac{1}{5}#, since the total area under the density curve has to equal #1#:

\[5\cdot x = 1 \Rightarrow x = \cfrac{1}{5}\]

The probability of event #A# is:

\[\mathbb{P}(A)=(5-3)\cdot \cfrac{1}{5}=\cfrac{2}{5}\] |

The probability of event #B# is:

\[\mathbb{P}(B)=(4-1)\cdot \cfrac{1}{5}=\cfrac{3}{5}\] |