Continuous Probability Models

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:

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

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