Question

Find the correlation p(X,Y), where X is the number of heads and Y is the number...

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?

Homework Answers

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.

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