Question

Find the correlation p(X,Y), where X is the number of heads and Y is the number of tails, if a biased coin is thrown with heads p and tossed n time?

Answer 1

Given X is the number of heads and Y is the number of tails, if a biased coin is thrown with heads p and tossed n times, then it has tails(n-p).

Let us imagine the given information in the form of the following table

Tossing number	Result	X	Y	XY	X2	Y2
1	H	1	0	0	1	0
2	H	1	0	0	1	0
.	.	.	.	.	.	.
.	.	.	.	.	.	.
p	H	1	0	0	1	0
P+1	T	0	1	0	0	1
P+2	T	0	1	0	0	1
.	.	0	1	0	0	1
.	.	0	1	0	0	1
n	T	0	1	0	0	1
Sums		Sx=p	Sy=n-p	Sxy=0	Sxx=p	Syy=n-p

Now, we calculate N = n Sxy - Sx Sy = n*0 – p(n-p) = – p(n-p)

A= n Sxx - Sx2 = np – p2 = p(n-p)

B= n Syy – Sy2 =n(n- p) –(n- p)2 =(n-p)(n-n+p) = (n-p)p = p(n-p)

D= SQRT(A*B)= SQRT(p(n-p)* p(n-p))= p(n-p)

Therefore, the correlation

p(X,Y) = N/D = – p(n-p)/ p(n-p) = – 1.