2. Answer the following questions. Prove your answers using derivations, following the properties of summation operator, expectations operator, and variance operator. Notations: (i) A random variable: X (ii) X has a distribution with mean, u, and variance, G. (iii) A random variable: Y (iv) Y has a distribution with mean, u, and variance, of- (v) A random sample of (X, Y ), with a sample size of n (vi) Sample mean of X's: x = _ 2 X i-=1 (vii) Sample mean of Y's: y = = Z y i-=1 (viii) Sample variance of X's: s = 1 2 X i-=1 (ix) Sample variance of Y 's: 5 = m-1 . 5. X i-=1 Suppose that the unit of measurement of x change as follows: (ex -*x). where c, b, k, and h are constants. How will the following change given this measurement change in X? a. Mean of X (H., or E(X)) b. Variance of X (o, or Var(X)) C. Covariance of X and Y (Cov(X, Y)) d. Correlation of X and Y (Corr(X, Y))