Chapter Objectives

Assignment

!! You’re not going to the HW section this time !!

  • All vocabulary (see Key Terms for definitions)
  • 4 Practice 1–3, 5–8, 10–17
    • Or if you prefer, 1–17 and skip 4 and 9
    • Solutions
  • Read the next section in the book

Probability distribution functions

You might remember function notation from Algebra class, where you had a function like $f(x) = 4x + 5$ and if you plugged in a number for $x$ it would produce an output. For example $f(2)=13$ because $4(2)+5=13$.

Probability distribution functions (or PDFs because that’s too long to type out repeatedly) work the same way, except we are going to only focus on the $f(2)=13$ part. We will completely ignore how we got from $f(2)$ to $13$. Here is an example of that in action.

$x$ $P(x)$
$0$ $\frac{2}{50}$
$1$ $\frac{11}{50}$
$2$ $\frac{23}{50}$
$3$ $\frac{9}{50}$
$4$ $\frac{4}{50}$
$5$ $\frac{1}{50}$

This is a PDF for for the number of times a newborn baby’s crying wakes a mother after midnight. A sample of 50 mothers was used to collect the data above. We have our input $x$ and our output $P(x)$, which is all we need to do some simple statistical analysis. What is ${P(x=2)}$? That’s just $\frac{23}{50}$. What about ${P(x\le2)}$? That’s $\frac{23}{50} + \frac{11}{50} + \frac{2}{50} = \frac{36}{50}$.

Discrete random variables

We talked about discrete versus continuous way back in 1.2, so I’ll be lazy and copy and paste my slide bullets.

  • Discrete quantitative data means only certain numbers, typically whole numbers
  • e.g., number of people in a household
  • Continuous quantitative data is data where all numbers in a range are valid
  • e.g., height, weight, time

The same applies to discrete and continuous random variables. The values are countable for a discrete one, and uncountable for a continuous. The table above is for a discrete variable since we can count the number of times each mother was torn from a peaceful slumber.

When dealing with discrete probability distribution functions, there are two other key points:

  1. Each probability is between zero and one, inclusive.
  2. The sum of the probabilities is one.