Question

Assume that a coin is flipped where the probability of coin lands "Heads" is 0.49. The coin is flipped once more. This time, the probability of obtaining the first flip's result is 0.38. The random variable X is defined as the total number of heads observed in two flips. On the other hand, the random variable Y is defined as the absolute difference between the total number of heads and the total number of tails observed in two flips. Calculate the correlation coefficient between X and Y.

Answer 1

Solution Given P(H) = 0.49 P/H1H) = 0.38 P(TIH) = (-0.38 P(TIH) = 0.62 Now the values of x,y along with their probabilities for different Combination of tosses here Result y p(xxx, x-y) HH Q 2 0.49 X 0.38 - 0.1862 0.3038 0.49 X 0.62 0.51 x 0.62 0.3162 TH 0.1938 0.51 X 0.38 TT Now to find E(X) E(x) = 2(0.1862) + (0.3638 +0.3162) E(X) E(*) E(x) 0.3724 +1 (0.62) 0.3724 + . 0.62 0.9924

= Now to find E(y) E(Y) = 2 x (0.1862 + 0.1938) 2 x 0.38 E(4) 0.76 Now to find E(x²) E(X) 2 x 0.186 2 + 1 2 (0.3 0. 3038 + 0.3162) 4 x 0.18624 (0.62) 0.74 48 + 0.62 E(x²) 1.3648 Now to find E(42) E(72) = 2 ? x (0.186.2 + 0.1938) = 4 X 0.38 E(42) = 1.52 Now to find Var(x) Var (x) F(x2) - [E (x} = 1,3648 - (0.9924) ? 1.3648 - 0.98486 War(x) = 0.3799 2

Now to find Varly) Var(y) = E(+2) - [[(y)] ² Varly) = 1.52-(0.76² = 1.52 -0.5776 Var (4) 0.9424 Now to find Elxy) E(XY) = 2x2xP(x=2, y=2) X 0.1862 E(xy) 0.7448 Cov(x,y) = E(XY) – E(X) Ely) 0.7648 - 66.9924) (0.76) - 0.7448 - 0.7542 [Cov(XY) =-0.0094

Now 4 to find the correlation between X andy Cor (x, y) pa var(x) Var(y) -0.0094 6.3799 (0.9924) -0.0094 = P 0.3770 -0.0094 으 O. 6140 - 0.0153 Therefore the correlation b. Coefficient between X and y is -0.0153