chap5S2

# 072401625011 464 thecovarianceis covx

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Unformatted text preview: random variables. A measure of the strength of a linear relationship between random variables X and Y is the correlation defined as: ρ = Corr(X , Y ) = Cov(X , Y ) σX σY ↑ Greek letter rho where σ X and σ Y are the standard deviations of the random variables. A result is: − 1 ≤ ρ ≤ 1 A value of ρ = 0 indicates that the random variables are uncorrelated. Problem: If two random variables are uncorrelated, are they independent ? 32 Chapter 5 Example: the real estate agent exercise Continued. Earlier in the lecture notes, an exercise introduced the joint probability function: X Y 0 1 2 3 0.09 0.14 0.07 4 0.07 0.23 0.16 5 0.03 0.10 0.11 To find the covariance between the random variables X and Y first calculate: E(X Y ) = ∑ ∑ x y PX , Y (x , y ) xy = (0)(3)(0.09) + (0)(4)(0.07) + (0)(5)(0.03) + (1)(3)(0.14) + (1)(4)(0.23) + (1)(5)(0.10) + (2)(3)(0.07) + (2)(4)(0.16) + (2)(5)(0.11) = 4.64 The covariance is: Cov(X , Y ) = E(XY ) − μ X μ Y = 4.64 – (1.15)(3.94) = 0.109 33 Chapter 5 The variances of the two random variables are calculated as: σ 2 = E(X 2 ) − μ 2 X X = (0)(0.19) + (1)(0.47) + (4)(0.34) – (1.15)(1.15) = 0.5075 σ 2 = E( Y 2 ) − μ 2 Y Y = (9)(0.30) + (16)(0.46) + (25)(0.24) – (3.94)(3.94) = 0.5364 The correlation between X and Y is: ρ= Cov(X , Y ) σ2 σ2 XY = 0....
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