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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