Final answer:
To compute Cov(X,Y) and Corr(X,Y) for the number of heads and tails after flipping four fair coins, one needs to take into account that there is a deterministic linear relationship between X and Y. Cov(X,Y) is negative due to this inverse relationship and the correlation will be -1. Furthermore, for independent random variables Y and Z, the correlation is always zero.
Step-by-step explanation:
The question asks to compute the covariance (Cov) and the correlation (Corr) between the random variables X, the number of heads, and Y, the number of tails, when four fair coins are flipped. It also asks to prove that if Y and Z are two independent random variables, their correlation is zero.
Computation of Cov(X,Y)
Since there are a total of four coins and X is the number of heads, for every head that occurs, there is one less tail. Therefore, Y can be determined directly from X by the relationship Y = 4 - X. Because X and Y are linearly related with negative slope, Cov(X, Y) would be negative. However, knowing the exact value requires calculations based on the probability distribution of X and Y.
Computation of Corr(X,Y)
The correlation coefficient is calculated as Corr(X, Y) = Cov(X, Y) / (σX*σY), where σX and σY are the standard deviations of X and Y. Given the deterministic inverse relationship between X and Y in this specific context, Corr(X, Y) would equal -1.
Proof of Corr(Y,Z)=0 when Y and Z are independent
To prove that Corr(Y, Z) = 0, we can use the definition of correlation and the fact that the covariance of two independent random variables is 0. If Y and Z are independent, then Cov(Y, Z) = 0, and by definition Corr(Y, Z) = Cov(Y, Z) / (σY*σZ). Since the covariance is zero, the correlation must also be zero. This is in keeping with the concept that independent random variables have no linear relationship.