If cov(X, Y)=1.0, what can we say about the relationship...

If $\operatorname{cov}(X, Y)=1.0$, what can we say about the relationship between $X$ and $Y$ ? If $\operatorname{cov}(Z, W)=2.0$, does this mean that the relationship between $Z$ and $W$ is stronger than the relationship between $X$ and $Y$ ? Explain.

Key Concepts

Correlation Coefficient

The correlation coefficient standardizes covariance by dividing it by the product of the standard deviations of the two variables, making it a dimensionless measure that ranges between -1 and 1. This standardized metric is more useful for comparing the strength and direction of relationships between different pairs of variables.

Scale Dependence

Scale dependence refers to the fact that covariance values are affected by the units in which the variables are measured. A higher covariance value does not necessarily indicate a stronger relationship unless the variables are on the same scale; hence, without standardizing the variables, comparisons of covariance across different pairs can be misleading.

Covariance

Covariance is a measure of the linear relationship between two random variables. It provides an indication of whether increases in one variable tend to be accompanied by increases in the other (positive covariance) or decreases in the other (negative covariance), but its magnitude is influenced by the scale of the variables, making it difficult to interpret the strength of the relationship directly.

Transcript

00:02 For part a, start from the definition, the covariance between x1 and x2 is equal to the expected value of the difference between x1 and mu 1 times x2 minus mu2.

00:21 That would be equal to the expected value of x1 times x2 minus mu 1 times x2 minus mu 2 times x1 plus mu 1 times x1, plus mu 1 times mu 2, which would be the expectant value of the expectant.

00:37 Value of x1 x2 minus mu 1 times the expected value of x2 minus mu 2 2 times the expected value of x1 plus mu 1 mu 2 in with mu 1 being the expected value of x1 and 2 being the expected value of x2 and the covariance would be e of x1 x2 minus e of x1 times e of x2.

01:11 For part b, use the variance of z being equal to the expected value of z squared minus the expected value of z squared, whereas z is equal to ax1 plus bx2 and then expand z2 and group the terms.

01:27 So if z is ax1 plus bx2, then z squared is a squared x1 plus bx2, then z squared is a squared x1 squared plus b squared x2 squared plus 2 a b x1 x2 e of z squared is equal to a squared e of x1 squared plus b squared e of x2 squared plus 2 a b b e of x1 x2 e of z is equal to a i i'll write it e of z is going to be a e of x1 plus b e of x2 which means that e of z squared would be a squared, e of x1 squared, plus b squared, e of x2 squared, plus 2, a, b of x1, plus 2, a, b of x1 times e of x2.

02:33 Subtract, you know, we get the variance of z being equal to e of z squared minus e of z squared, being equal to a squared times e of x1 squared minus e of x1 squared plus b squared times e of x2 squared, plus b squared times e of x2 squared, minus e of x2 squared, plus 2 ab times e of x1 x2.

03:16 So that would mean that the variance of a x1 plus b x2 is equal to a squared times the variance of x1 plus b squared times the variance of x2 plus two a b times the covariance of x1 x2 for part c with covariance the variance gets a cross term minimizing gives a formula for the best k and it equals 0 .5 only in special cases.

03:56 So the variance of x would be equal to k squared, sigma 1 squared, plus 1 minus k squared, sigma 2 squared plus 2k times 1 minus k times sigma sub 1 2, where sigma 1 squared is the variance of x 1, sigma 2 squared is variance of x2 and sigma 1 2 is the covariance between x1 and x2.

04:22 So if we differentiate and set equal to 0, d d d k of the variance so x would be 2k, sigma 1 squared minus 2 times 1 minus k, sigma 2 squared plus 2 times 1 minus 2k, sigma sum 1 2.

04:40 When we set that equal to 0 and solve, so k would be equal to sigma 2 squared minus sigma 1 2 over sigma 1 squared plus sigma 2 squared minus 2 sigma 1 2.

04:54 Again, that's the covariance.

04:56 So it's not always 0 .5 for part d.

05:05 Correlation is a standardized covariance...

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