Continuous distribution

A continuous variable is, for example, age or time or even the area of an apartment. Such variables take on an infinite number of values, and not all possible values can be listed. The closer the issue can be measured, the more accurate the values of the variable will be. Can you list all possible real numbers between the numbers 0 and 1? If you choose any two adjacent numbers, there are always numbers between them.

Probability density function

A continuous random variable cannot be represented in a table, so it is usually represented using a density function.

The area of the area bounded by the function and the x-axis between [a, b] is equal to the probability in this interval.


The total probability is always 1, so the area bounded by the density function and the x-axis is always 1.

The density function has the following conditions.


  • The function is always positive for all values of x.

  • The area bounded by the graph of the function and the x-axis is 1.

  • The probability between [a, b] is always equal to the area of the corresponding region.

Example 1

We define the function f(x) as follows

Find the constant a such that the function f is a density function.

The function is always positive, so it must be set that its area between [0,2] is 1.

Integrate

Find a, when are is 1

Cumulative distribution function

The distribution can also be described by the cumulative distribution function. The value of this function indicates the amount of probability accumulated so far.

The cumulative distribution function is obtained from the density function

Example 2

Find the cumulative distribution function for the density function of Example 1. Since the function is 0 when x is less than zero, it can be integrated between [0, t]

Let's change to the variable x and since all probabilities have accumulated at t = 2, we get the cumulative distribution function

Example 3

The density function of a random variable x is

a) Draw a graph of the density function.

b) Calculate the probabilities P (x ≤ 1), P (1 <x ≤ 3) and P (x> 3)

(YO2003S Long Mathematics)

a) we use geogebra to draw the graph

b) Find the probabilities P (x ≤ 1), P (1 <x ≤ 3) and P (x> 3)

The graph is relatively easy, so there is no need to integrate. We can find the areas using triangles.

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