Solution:

Given, a coin is tossed n times and X and Y denote the number of heads and tails turned up respectively.

To find: The correlation coefficient between X and Y, r(X,Y).

Formula used: r(X,Y) = [(n∑XY) - (∑X∑Y)] / √[(n∑X^2 - (∑X)^2)(n∑Y^2 - (∑Y)^2)]

where ∑ represents the sum of the variable over n trials.

Calculation:

Let us consider the possible outcomes of tossing a coin n times:

- Head (H)
- Tail (T)

The total number of possible outcomes = 2^n

Let us represent the number of heads and tails using X and Y respectively.

Therefore, X + Y = n

The possible values of X and Y are:

- If X = 0, Y = n
- If X = 1, Y = n-1
- If X = 2, Y = n-2
- .
- .
- .
- If X = n, Y = 0

We can represent the possible outcomes in a tabular form as shown below:

Outcome | X | Y | XY | X^2 | Y^2
--------|---|---|----|-----|-----
1 | 0 | n | 0 | 0 | n^2
2 | 1 | n-1| 1 | 1 | (n-1)^2
3 | 2 | n-2| 2 | 4 | (n-2)^2
. | . | . | . | . | .
. | . | . | . | . | .
. | . | . | . | . | .
n+1 | n | 0 | 0 | n^2 | 0

Using the above table, we can find the values of ∑X, ∑Y, ∑XY, ∑X^2, and ∑Y^2.

- ∑X = 0+1+2+...+n = n(n+1)/2
- ∑Y = n(n+1)/2
- ∑XY = 0+1* (n-1)+2* (n-2)+...+n*0 = n(n-1)/2
- ∑X^2 = 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6
- ∑Y^2 = n(n+1)(2n+1)/6

Substituting the above values in the formula for r(X,Y), we get:

r(X,Y) = [(n*n(n+1)(n-1)) / 2 - (n(n+1)/2 * n(n+1)/2)] / √[(n(n+1)(2n+1)/6 - (n(n+1)/2)^2)(n(n+1)(2n+1)/6 - (n(n+1)/2)^2)]

On simplifying, we get:

r(X,Y) = -1

Therefore