Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

The solutions for problem 5-67 in the stat 345 (probability and statistics) textbook, third edition. The problem involves finding the covariance and correlation between two random variables x and y based on their marginal distributions. The document walks through the calculations step by step, starting with finding the expected values e(xy), e(x), and e(y), then calculating the variances var(x) and var(y), and finally determining the covariance and correlation.

Typology: Study notes

Pre 2010

1 / 2

Download Solutions for Stat 345 Problem 5-67: Covariance & Correlation of Two Variables and more Study notes Mathematical Statistics in PDF only on Docsity! Stat 345 Solutions - Section 5.5 (3rd edition) Problem 5-67 First, we find the marginal distributions. The marginal distribution for X is x 1 2 4 fX(x) 3 8 1 2 1 8 The marginal distribution for Y is y 3 4 5 6 fY (y) 1 8 1 4 1 2 1 8 The covariance is given by cov(X,Y ) = E(XY )โ E(X)E(Y ) First, we find E(XY ): E(XY ) = โ x โ y xyfXY (x, y) = (1)(3) 1 8 + (1)(4) 1 4 + (2)(5) 1 2 + (4)(6) 1 8 = 9.375 Now we find E(X) and E(Y ): E(X) = (1)3 8 + (2)1 2 + (4)1 8 = 1.875 E(Y ) = (3)1 8 + (4)1 4 + (5)1 2 + (6)1 8 = 4.625 Thus, cov(X, Y ) = 9.375โ (1.875)(4.625) = 0.7031. The correlation is given by corr(X, Y ) = cov(X, Y )โ V ar(x)V arY So we need to find the variances: V ar(X) = E(X2)โ E(X)2 = (1)3 8 + (4)1 2 + (16)1 8 โ 1.8752 = 0.8594 V ar(Y ) = E(Y 2)โ E(Y )2 = (9)1 8 + (16)1 4 + (25)1 2 + (36)1 8 โ 4.6252 = 0.7344