=B1. Let X₁ and Y; be random variables with Var(X;) = o² and Var(Y₂) o for allie {1,...,n}. Assume that each pair (X₁, Y;) has correlation Corr(X, Y) = p, butthat (X₁, Y₂) and (X₁, Y;) are independent for all i j.(a) What is Cov(X₁, Yi) in terms of Ox, Oy(b) Show that Cov(X₁,Y) = poxoy/n, where Y is the average of the Y₁.(c) Determine Cov(X, Y).and p?

(a) The covariance between Xi and Yi is given by: Cov(Xi, Yi) = E[(Xi - E[Xi])(Yi - E[Yi])] Since…

Answered: Let X and Y be random variables with…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 52RE

See similar textbooks

Similar questions

EXER 6.3 Find the covariance and the correlation coefficient between X and Y, if X and Y are jointly discrete random variables, with joint PMF given by: SHOW SOLUTIONS X\Y 0 1 6 0 28 6 1 28 2 0 333333 28 28 28 2120 28 0
Let Xi and Yi be random variables with Var(Xi) = σx2 and Var(Yi) = σy2 for all i ∈ {1, . . . , n}. Assume that each pair (Xi, Yi) has correlation Corr(Xi, Yi) = ρ, but that (Xi,Yi) and (Xj,Yj) are independent for all i ̸= j. (a) What is Cov(Xi,Yi) in terms of σx, σy and ρ? (b) Show that Cov(Xi,Y ̄) = (ρσxσy)/n, where Y ̄ is the average of the Yi (c) Determine Cov(X ̄,Y ̄). B2. Consider the random variables Xi and Yi from question B1 again. (a) Show that the sample covariance is an unbiased estimator of Cov(X1,Y1). Hint: consider the equality Xi − X ̄ = (Xi − μ) − (X ̄ − μ). (b) Can you conclude from the statement in part (a) that the sample correlation is an unbiased estimator of Corr(X1, Y1)? Justify your answer.
The correlation between X and Y Select one or more: a. is the covariance squared b. cannot be negative since variances are always positive. c. is given by corr(X, Y) = cov(X,Y)/var(X)var(Y)cov(X,Y)/var(X)var(Y) d. can be calculated by dividing the covariance between X and Y by the product of the two standard deviations
Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of Y
C2. Let X and Y be random variables, and let a and b be constants. (a) Starting from the definition of covariance, show that Cov(aX, Y): = a Cov(X, Y). You may find it helpful to remember that if EX = µx, then EaX = αμχ· (b) Show that Cov(X + b, Y) = Cov(X, Y). Now let X, Y, Z be independent random variables with common variance o². (c) Find the value of Corr(2X - 3Y + 4, 2Y – Z - 1). You may use any facts about covariance from the notes, including those from parts (a) and (b) of this question, provided you state them clearly.
Let X and Y have joint PDFfX,Y (x, y) = 3x^2y + 3xy^2 if 0 < x < 1, and 0 < y < 1 0 otherwise.a) Find the covariance of X and Y .b) Find Var(X).c) Find the correlation of X and Y (X and Y have the same variance.)
Let X1, X2, and.X3 be independent and normally distributed random variables with E(X1) 4, E(X2) = 3, E(X3) = 2, Var(X1) = 1, Var(X2) = 5, Var(X3) = 2. Let Y = 2X1 + X2 – 3X3. Find 2. the distribution of Y.
Let X and Y be independent uniform random variables on (0,1). a) Find the variance of Z1=min(X,Y). b) Find the variance of Z2 = max(X,Y).
A) Caused by a third variable B) uncorrelated ) negative effect 0.fb/ orrrte D) direct cause-and-effect 7) The formula for the correlation coefficient r= n (Σxy)-(Σx) (Σν) means 7) n(Ex2)- (Ex)2n(2y2)- (2y) D A) r is the ratio of the covariance to the product of the standard deviation of XY B) r is the ratio of the covariation to the square root of the product of the variation in X and the variation in Y. C)ris the ratio of the covariance to the square root of the product of the variation in X and the variation in Y. D) r is the ratio of the covariation to the product of the standard deviations of X and Y.
If X is a random variable, prove that Cov(X,X) = σX².
Let X1, X2, · · · , Xn be uniform i.i.d. random variables between [0, 1].Find the CDF and PDF of X = min(X1, X2, · · · , Xn)
2. Let X and Y be random variables, and let a and b be constants. (a) Starting from the definition of covariance, show that Cov(aX,Y) = a Cov(X,Y). You may find it helpful to remember that if EX = µx, then EaX = aux. (b) Show that Cov(X + b, Y) = Cov(X,Y). Now let X, Y, Z be independent random variables with common variance o?. (c) Find the value of Corr(2X – 3Y + 4, 2Y – Z – 1). You may use any facts about covariance from - the notes, including those from parts (a) and (b) of this question, provided you state them clearly.

SEE MORE QUESTIONS

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

Let X and Y be random variables with Var (X₂) o and Var(Y₂) o for all ie {1,...,n}. Assume that each pair (X₁, Yi) has correlation Corr(X₁, Y) = p, but that (X₁, Y₂) and (X, Y,) are independent for all i j. = =