Continuous Probability Distributions

The p.d.f. of a continuous random variable is:

(see definite integral)

can take on values larger than 1, but the area under must equal 1 (recall the definition of an integral). For example, if at , and everywhere else, then the area underneath is still .

What does mean?

Take :

so you have the probability from to , which means we're looking at a window of size (that is, a rectangle centred at of width ).
The approximation to means that is the width of the rectangle, and is the height of the rectangle at point (i.e the area of said rectangle).
Taking the limit: .

This sort of resembles the epsilon - delta stuff.

If and , then is roughly twice as likely to occur as (remember, it's an approximation which gets more accurate as ).

Let be a random variable with the following p.d.f.:

left=-1; right=2;
top=3; bottom=-1;
---
3x^2  \{0 \le x \le 1\}
0  \{x \lt 0, x \gt 1 \}
(1, 0) | open | #388c46

1. Is it a real p.d.f.?
   Is ? Yes.
   Does it add to 1?

Yes. Real.

2. Find the c.d.f.:

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

3. Find :

Another way we could've done this: take the differences in the c.d.f (which would also be .

4. Find the expected value:

5. Find

6. Find