**Problem 6-66**Let \( x \) and \( y \) be independent random variables with variances \( \sigma_1^2 \) and \( \sigma_2^2 \), respectively. Consider the sum\[z = ax + (1-a)y \quad \text{where} \quad 0 \leq a \leq 1\]Find \( a \) that minimizes the variance of \( z \).

Answered: Image /qna-images/answer/8d1e35e8-a66d-4bd3-951e-4c0277f65328.jpg

Answered: 6-66 Let x and y be independent random…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

A certain automobile manufacturer equips a particular model with either a turbo or a standard engine. Let X1 and X2 be fuel efficiencies for independently and randomly selected turbo and standard cars, respectively. Assume the two random variables are normally distributed with the following parameters: H = 22 and o,2 =1.44 (mean and variance of X1] Hi = 26 and = 2.25 [mean and variance of X2] Define a random variable Y as the difference between the first engine minus three times the second engine, X, - 3X, Find V(Y), the variance of the linear combination Y
3. Suppose X1 and X2 are independent random variables with variances o? > 0 and o? > 0, respectively. Let U = X1 and V = X1+ X2. Find p(U, V) in terms of ơ1 and 02.
Prove that the variance of a beta-distributed random variable with parameters a and ß is αβ (a + B)² (a + B + 1)* =
3. Let X1, X2, ..., Xn be a random sample from the population N(u, o²) and Y1, Y2,...,Ym a random sample from the population N(uy, o²), where both means (ua and uy) are assumed known. Note that the two distributions have a common variance of o?. (a) If the X's and Y's are independent, show that the MLE for the common variance is: E(Xi - Ha)² +E(Y; - Hy)² n + m (b) Is the MLE found above unbiased for o??
Find variance of Y1
show that if X11, X12,..., X1, X21, X22,..., X2n₂ are independent random variables, with the first n constitut- ing a random sample from an infinite population with the mean μ₁ and the variance of and the other n2 constitut- ing a random sample from an infinite population with the mean μ2 and the variance o2, then 1 M2 (a) E(X₁-X₂) = μ₁ −μ2; 07 0 22 (b) var(X₁-X₂) = + n₁ n₂
We have the following information about the random variables X and Y: |x = 1, |y = −1.5 , ox = 0.25, c} = 1. o Calculate the variance of Z = 3X + 5Y, a) when the coefficient of correlation is p(X, Y) = 0.46: 0²77 34.15 Z b) when X and Y are independent random variables: 0²2 27.75 X
Let X and Y be independent random variables with μX = 3, σX = 3, μY = 2, and σY = 3. Find the mean and variance of X – Y. The mean of X - Y is . The variance of X - Y is .
(b) Let X₁, X₂, X3 be uncorrelated random variables, having the same variance ². Consider the linear transformations Y₁ = X₁ + X₂, Y₂ = X₁ + X3 and Y3 = X₂ + X3 . Find the correlations of Yi, Y; for i #j. (5 marks)
Assume that X and Y are random variables with the following parameters: 4 аnd o'y 13, 12, о Hy What is the variance of Z if Z = 2X - Y + 1 and ory =0.8
R3

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

6-66 Let x and y be independent random variables with variances of and o2, respectively. Consider the sum z = ax + (1 − a)y Find a that minimizes the variance of z. Osasl