Cumulative distribution function

Imagine you're at a store with a bunch of candies in front of you. You look at each candy and decide how much you want to pay for it. Let's say you're willing to pay up to $1 for the first candy, $2 for the second candy, and $3 for the third candy.

We can write this information down in a table:

| Candy number | Maximum price |
|--------------|---------------|
| 1 | $1 |
| 2 | $2 |
| 3 | $3 |

Now, let's say a friend of yours comes to the store and also wants to buy some candy. They say that they're willing to pay up to $2 for the first candy, $3 for the second candy, and $4 for the third candy.

We can add this information to the table:

| Candy number | Your maximum price | Friend's maximum price |
|--------------|--------------------|------------------------|
| 1 | $1 | $2 |
| 2 | $2 | $3 |
| 3 | $3 | $4 |

Now, let's think about a different question: what is the most you and your friend would be willing to pay for any candy? We can answer this question by looking at the cumulative distribution function, which is a fancy way of saying "how much money you and your friend would be willing to spend in total on all the candies up to a certain point".

To create the cumulative distribution function, we add up the maximum prices for you and your friend at each candy number. For example, to find the total you and your friend would be willing to spend on the first candy, we add up $1 (your maximum price) and $2 (your friend's maximum price), giving us $3.

We can write this information down in another table:

| Candy number | Maximum total price |
|--------------|---------------------|
| 1 | $3 |
| 2 | $5 |
| 3 | $7 |

This table shows the cumulative distribution function: for each candy number, it tells us the maximum amount you and your friend would be willing to spend on all the candies up to and including that candy.

So, to sum up: the cumulative distribution function tells us how much money you and your friend would be willing to spend on all the candies up to a certain point. It's like a running total of the maximum prices you and your friend would pay for each candy, added up as we go along.