(5) Show Corr(aX + b, cY + d) = Corr(X, Y). Hint: Use results for the covariance and variance.] (5) Show Corr(aX + b, cY + d) = Corr(X, Y). Hint: Use results for the covariance and variance.]
(5) Show Corr(aX + b, cY + d) = Corr(X, Y). Hint: Use results for the covariance and variance.]
1. Show that Corr(aX + b, cY + d) = Corr(X, Y) using the definition of correlation in page 249, and finding each component first. Correlatiorn DEFINITION The correlation coefficient of X and Y, denoted by Corr(X, Y), or ρχ.r, or just ρ, is defined by Ox-GY
Prove the following statements • corr(ax,y) = corr(x,y) • show that if x,y and z are independent. Show what happened to: cov(x+y,x+z)= ? • assume x and y are not independent: cov(ax + b, y)= ? 70 tre la Car
Show that if Y = ax + b (a = 0), then Corr(X, Y = +1 or -1. We know Cov(X, Y) = Covl X, a X ) +o) - (1 ])uxo. X). Then Cov(X, Y) oxor Jux Corr(x, y) = which is 1 when a > 0 and –1 when a < 0 0x (lal ox) lal Under what condition will = +1? The value p = +1 when a > 0
Prove the following properties using the definition of the variance and the covariance: Q1. Operations with expectation and covariances Recall that the variance of randon variable X is defined as Var(X) Ξ E [X-E(X))2], the covariance is Cov(X, ) EX E(X))Y EY) As a hint, we can prove Cov(aX + b, cY)-ac Cov(X, Y) by ac EX -E(X)HY -E(Y)ac Cov(X, Y) In a similar manner, prove the following properties using the definition of the variance and the covariance: (a) Var(X)-Cov(X,...
Ex. 36Show that if Y= aX+ b (a≠0), then Corr(X, Y) = +1 or -1. Under what conditions will p = +1?
Let X and Y b Var(Y) (1) If a, b,c and d are fixed real numbers, = E(X), μγ E (Y),咳= Var(X) and e ranclom variables. with y a) show Cov(aX +b, cY +d)- ac Cov(X,Y) (b) show Corr(aX + b, cY + d)-PXY for a > 0 and c > 0.
Exercise 1 (1). X, Y are random variables (r.v.) and a,b,c,d are values. Complete the formulas using the expectations E(X), E(Y), variances Var(X), Var(Y) and covariance Cov(X, Y) (a) E(aX c) (b) Var(aX + c (d) Var(aX bY c) (e) The covariance between aX +c and bY +d, that is, Cov(aX +c,bY +d) f) The correlation between X, Y that is, Corr(X,Y (g) The correlation between aX +c and bY +d, that is, Corr(aX + c, bY +d)
C5) If we have two planes i : AX + B y + Cy + D, = 0 and 1: A x + B x + C + D = Othat are parallel but not coincident, what must be true of the coefficients A1, A2, B1,B2, C1,C2, D1,D2? (2 marks)
= Var(X) and σ, 1. Let X and Y be random variables, with μx = E(X), μY = E(Y), Var(Y). (1) If a, b, c and d are fixed real numbers, (a) show Cov (aX + b, cY + d) = ac Cov(X, Y). (b) show Corr(aX + b, cY +d) pxy for a > 0 and c> O