18: Probability - Covariance & Correlation

01 The Units Trap: covariance changes, correlation doesn’t

A researcher measures temperature \(T\) (°C) and ice-cream sales \(S\). They convert \(T\) to Fahrenheit:

\[F = 1.8T + 32.\]

    1. Using covariance properties, express \(\operatorname{Cov}(F,S)\) in terms of \(\operatorname{Cov}(T,S)\).
    1. Prove that the correlation is unchanged: \(\rho_{F,S}=\rho_{T,S}\), using the definition of correlation.
    1. Explain (2–4 sentences) why correlation is “unit-free,” while covariance is not.

02 Correlation \(=\pm 1\) as a detective test (constructive, not computational)

You are given \(x=[1,2,3,4,5,6]\).

    1. Construct integer-valued \(y\) such that the correlation is exactly \(+1\).
    1. Construct integer-valued \(y\) such that the correlation is exactly \(-1\).
    1. Justify both by referencing the condition for extremal correlation (the \(Y=aX+b\) characterization).

03 Dependent but uncorrelated (the “nonlinear dependence” trap)

Let \(X\) be uniform on \(\{-2,-1,1,2\}\), and define \(Y=X^2\).

    1. Compute \(\mathbb{E}[X]\), \(\mathbb{E}[Y]\), and \(\mathbb{E}[XY]\).
    1. Use \(\operatorname{Cov}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\,\mathbb{E}[Y]\) to show \(\operatorname{Cov}(X,Y)=0\).
    1. Explain why this does not mean independence (connect explicitly to the “correlation only measures linear dependence” warning).

04 Pearson vs Spearman: monotonic but not linear

Consider \(x=[1,2,3,4,5,6]\) and \(y=[1,2,4,8,16,32]\).

    1. Argue (without calculating Pearson exactly) why Pearson correlation is not \(1\) (use the “linear relationship” criterion).
    1. Compute Spearman rank correlation exactly (no ties here).
    1. One sentence: why Spearman is the right tool here.

05 One outlier can flip the story

Common points:

\[(1,1),(2,2),(3,3),(4,4),(5,5).\]

Dataset A adds \((6,6)\). Dataset B adds \((6,-20)\).

    1. For each dataset, decide whether the sample correlation is positive or negative (justify using the sign of “products of deviations,” not full computation).
    1. Explain how one point can dominate this “one-number summary.”

06 Diversification as a decision: when does combining reduce risk?

Two daily returns \(R_1, R_2\) satisfy:

\[\operatorname{Var}(R_1)=4,\quad \operatorname{Var}(R_2)=9,\quad \operatorname{Cov}(R_1,R_2)=c.\]

Let \(P=R_1+R_2\).

    1. Express \(\operatorname{Var}(P)\) as a function of \(c\).
    1. Compare \(c=+5,\,0,\,-5\). Which case yields the smallest portfolio variance, and why?
    1. Translate the result into plain English (what does negative covariance “do” for you?).

07 Correlation as geometry: angle between centered vectors

You are given mean-centered vectors:

\[u=(1,0,-1),\qquad v=(2,-1,-1).\]

    1. Compute \(\cos(\theta)=\dfrac{u\cdot v}{\lVert u\rVert\,\lVert v\rVert}\).
    1. Interpret the sign and magnitude of \(\cos(\theta)\) as a correlation analogue.
    1. Decide whether the variables are “roughly aligned,” “roughly orthogonal,” or “roughly opposite.”

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