1   Tutorial on probability density

Since the meaning of probability density when you transform variables is still causing problems for some people, think of changing units from English to metric. First, with one variable, X.

1. Let X be in feet and be U[0,1].

   $$f_X(x) = \begin{cases} 1& \text{if } 0\le x\le1\\ 0&\text{otherwise} \end{cases}$$

2. $P[.5\le x\le .51] = 0.01$.

3. Now change to centimeters. The transformation is $Y=30X$.

4. $$f_Y(y) = \begin{cases} 1/30 & \text{if } 0\le y\le30\\ 0&\text{otherwise} \end{cases}$$

5. Why is 1/30 reasonable?

6. First, the pdf has to integrate to 1: $$\int_{-\infty}^\infty f_Y(y) =1$$

7. Second, $$\begin{align} & P[.5\le x\le .51] \\ &= \int_.5^.51 f_X(x) dx \\& =0.01 \\& = P[15\le y\le 15.3] \\& = \int_{15}^{15.3} f_Y(y) dy \end{align}$$

2   Mathematica demo

1. Int
2. Sum
3. Manipulate
4. Binomial etc

3   Examples

4.11, p153.

4   4.3.2 Variance

p160

5   Memoryless Exponential Distn

p 166.

6   4.4.3 Normal (Gaussian) dist

p 167.

Show that the pdf integrates to 1.

Lots of different notations:

Generally, F(x) = P(X<=x).

For normal: that is called $\Psi(x)$ .

$Q(x) = 1-\Psi(x)$ .

Example 4.22 page 169.

7   4.4.4 Gamma r.v.

1. 2 parameters
2. Has several useful special cases, e.g., chi-squared and m-Erlang.
3. The sum of m exponential r.v. has the m-Erlang dist.
4. Example 4.24 page 172.

8   Functions of a r.v.

1. Example 4.29 page 175.
2. Linear function: Example 4.31 on page 176.

9   Comic

Dilbert