Analytics Basics and Continuous Probability Distributions

Recap: We're remembering what we learned before, like the basic ideas of checking data, what we expect to happen, how to guess what will happen using what we know, how likely different things are to happen, and how spread out our guesses might be.

We're starting with things that can only be certain separate amounts, like counting apples.

Discrete: These are things we can count, like how many fingers you have. You can't have half a finger!
- Example: If you draw a picture of how many kids have 1, 2, or 3 apples, you'll have separate bars for each number.
- Randomness: When you flip a coin, you don't know if it will be heads or tails, but you know it has a good chance of being either one.

Continuous: These are things that can be any amount, like how tall you are. You could be 4 feet and a tiny bit!

Imagine a pizza place that gives you a discount if the pizza is late (more than 30 minutes).
We want to guess how long the pizza will take so we don't have to pay discounts.
Pizza times could be 25 minutes and a little bit of a second, or 25 minutes and a different little bit of a second. It can be any time!
Even though we can only measure to the nearest second, there are so many possible times that it's easier to think of them as any possible time.

Pizza Delivery Time (PDT): Because we never know exactly how long the pizza will take, we call it a "random variable."
- If we write down all the pizza times, we'll see the average time and how much the times change.
- We can draw a picture (a "probability distribution") that shows us what times are most likely.
- To make the picture, we need to find a math rule (a "functional form") that fits the times.

Imagine a game where you win a prize between $0 and $10. Every amount is equally likely.

A pizza place always delivers between 20 and 40 minutes, and every time is equally likely.
Even though it's hard to measure exactly 35 minutes and 2 seconds, we pretend it's possible.

If we draw a line that connects all the possible pizza times, what's the area under the line?
The line is special because it shows us how likely each time is.

The area under the line adds up all the chances of having different pizza times.
Because the pizza has to be delivered sometime between 20 and 40 minutes, the area is 1 (or 100%).- $\sum p_i = 1$

If we have separate bars, each bar has a chance. But if we have a line, it's almost impossible for the pizza to be delivered in exactly 30 minutes.
Think of it like this: there are so many tiny times around 30 minutes (like 30 minutes and one small bit of a second, 30 minutes and two tiny bits of a second, etc.) that the chance of exactly 30 minutes is almost zero.- $p \approx 0$

Instead of asking about exactly 30 minutes, we ask: What's the chance the pizza will be here between 20 and 30 minutes?
This is like finding the area under the line between 20 and 30 minutes.
Another question: What's the chance the pizza will take more than 30 minutes?
- Because the whole area is 1, and the chance is the same for all times, it's 1/2 (or 50%).

When we have a line, the important thing is the area between two times, not the chance of any one time.

We can use lines to guess about things like pizza times, how much water flows from a tap, or how much rain falls.