Uniform Continuous Distribution

Definition

if we have p.d.f.:

We will often use because it is useful.

Intuition

This is just a uniform (same) property in a certain window

left=-1; right=4;
bottom=-1; top=4;
---
y = 1 \{1 \le x \le 2\}
y = 0 \{1 \ge x, x \ge 2 \}
(1, 0) | open | #388c46
(2, 0) | open | #388c46
(1, 1) | #2d70b3
(2, 1) | #2d70b3

The area under the line must be 1. You may also have a line from to , or to for example.

Intuition

If we had a rectangle with width , the only way for that rectangle to have area would be if its height was .

Support and CDF

(support and c.d.f.)

Support

CDF

Proof

(see integral)

Visualization

left=-1; right=3;
bottom=-1; top=2;
---
y = x - 1 \{1 \le x \le 2\}
y = 0 \{1 \ge x \}
y = 1 \{x \gt 2 \}

Expectation and Variance

(expected value and variance)

Definition

Given :

Proof

Explanation for (1): see LOTUS

Examples

Example

Let . Find .

solution

alternatively:

Notice how we ended up doing basically the same thing.

Universality

Suppose is a random variable with c.d.f. . If we are interested in generating simulations of (i.e we want to generate outcomes ). How do we do this?

We have where are values that come from a uniform distribution.

Notes:

needs to be invertible and "well-behaved"
A computer divides into a huge number of discrete chunks () and randomly select this list.

todo Slide 7, percentile illustration. Percentile spread should be uniform, with the same number of people in each percentile (not same score, e.g 70-79% and 80-80% both contain 10% of the population/class).

[!example]
Suppose has a pdf:

in general.

then

[!theorem]
Let be a continuous random variable with invertible c.d.f. . Then:

i.e when you plug a random random variable into its own CDF, you get a uniform distribution out of it.

In a way we're saying

[!proof]
Let .

Uniqueness: if a CDF or PMF behaves like an distribution, then it is that distribution.

So,

which is the same CDF as a uniform distribution. Therefore, must be .