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Solutions for Stat 345 Problem 5-67: Covariance & Correlation of Two Variables, Study notes of Mathematical Statistics

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

Uploaded on 07/23/2009

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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 )โˆ’ E(X)E(Y ) First, we find E(XY ): E(XY ) = โˆ‘ x โˆ‘ 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 ) = 9.375โˆ’ (1.875)(4.625) = 0.7031. The correlation is given by corr(X, Y ) = cov(X, Y )โˆš V ar(x)V arY So we need to find the variances: V ar(X) = E(X2)โˆ’ E(X)2 = (1)3 8 + (4)1 2 + (16)1 8 โˆ’ 1.8752 = 0.8594 V ar(Y ) = E(Y 2)โˆ’ E(Y )2 = (9)1 8 + (16)1 4 + (25)1 2 + (36)1 8 โˆ’ 4.6252 = 0.7344