Statistics and Probability - tossing a coin

Question

Two fair coins are to be tossed and the outcome is the ordered pair (face on the first coin, face on the second coin). Let the face “tail” be denoted by T and let the face “head” be denoted by H.

Let X be the number of heads in the toss of two coins

Let Y be number of tails in the toss of two coins.

Find the correlation coefficient of X and Y . (Hint: Y = 2 − X.)

Answer 1

Results are 1/4HH 1/2HT or TH and 1/4TT x(i) = (2,1,0) y(i) = 0,1,2

x_avg = sum of p(x_i)x_i = 1/4(2) + 1/2(1) + 1/4(0) = 1 same for y_avg = 1

The correlation coefficient is the covariance of x and y divided by the product of the standard deviations of x and y.

Covariance is E((x-x_avg)(y-y_avg) = (1/4)(2-1)(0-1) + (1/2)1-1)(1-1) + 1/4(0-1)(2-1) = -1/2

The variance of x is E((x-x_avg)²) = 1/4(2-1)² + 1/2(1-1)² + 1/4(0-1)² = 1/2 (same for y)

This leads to σ_xσ_y = (1/sqrt(2))(1/sqrt(2)) = 1/2

So correlation coefficient is (-1/2)/(1/2) = -1

Beyond this the hint is telling you that the line of best fit is y = 2-x for which the points are completely inversely correlated r = -1

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