Two fair coins are tossed independently. X is a random variable that takes a value of 1 if both tosses are heads and 0 otherwise. Y is a random variable that takes a value of 1 if at least one of the tosses is heads and 0 otherwise. The value of the covariance of X and Y is __________ (rounded off to three decimal places).

GATE 2024 · Engineering Mathematics · Random Variable · medium

Answer: Cov(X, Y) = 1/16 = 0.0625 (approximately 0.062 to three decimal places)

1. Identify outcome probabilities: 4 equally likely outcomes: HH, HT, TH, TT each with probability 1/4. X=1 only at HH. Y=1 at HH, HT, TH.
2. Compute E[X] and E[Y]: E[X] = P(HH) = 1/4. E[Y] = P(HH or HT or TH) = 3/4.
3. Compute E[XY]: XY = 1 only at HH (X=1, Y=1). At all other outcomes XY = 0. So E[XY] = 1*(1/4) = 1/4.
4. Apply covariance formula: Cov(X,Y) = 1/4 - (1/4)*(3/4) = 4/16 - 3/16 = 1/16 = 0.0625.