Thanks ]]>

(2-1), (3-1), ... (59-1)

(3-2), (4-2) ... (59-2)

.

.

(58-57), (59-57)

(59-58)

Explain in simple words please

Hi,

Let's begin with the list of |i - j| possibilities :

1, 2, 3, 4, ... , 58

1, 2, 3, 4 .. 57

1, 2, 3, .. 56

.

.

.

1, 2

1

Each number from every row is equiprobable

Hence, the expectation:

and for any n

This is the answer from my notes.

There are n(n-1) ways to pick two numbers from the unit interval 1 ... n.

If they are ordered high and low then there are:

ways.

The standard way now to do this is to look at the ordered pairs. There is 1 way for 2 to be the maximum and 2 ways for 3 to be the maximum and there are generally m-1 ways for m to be the maximum. Using the formula for expected value or expected number:

For the second part, you do it in the same way. There are

still n (n-1)/2 ways to arrange n numbers with high and low.

Now you go through the numbers in the same way. 1 is the lowest n-1 times. 2 is the lowest n - 2 times. n can never be the lowest. n-1 is the lowest one time so generally m is lowest n - m times. We use the same formula for expected value:

Now for the difference.

Just subtract the expected value of the lowest from the highest.

The absolute value is implied in there.

When you have time please sign up over here:

]]>]]>