Paper 2, Section I, F

Probability | Part IA, 2010

Let XX and YY be two non-constant random variables with finite variances. The correlation coefficient ρ(X,Y)\rho(X, Y) is defined by

ρ(X,Y)=E[(XEX)(YEY)](VarX)1/2(VarY)1/2\rho(X, Y)=\frac{\mathbb{E}[(X-\mathbb{E} X)(Y-\mathbb{E} Y)]}{(\operatorname{Var} X)^{1 / 2}(\operatorname{Var} Y)^{1 / 2}}

(a) Using the Cauchy-Schwarz inequality or otherwise, prove that

1ρ(X,Y)1-1 \leqslant \rho(X, Y) \leqslant 1

(b) What can be said about the relationship between XX and YY when either (i) ρ(X,Y)=0\rho(X, Y)=0 or (ii) ρ(X,Y)=1|\rho(X, Y)|=1. [Proofs are not required.]

(c) Take 0r10 \leqslant r \leqslant 1 and let X,XX, X^{\prime} be independent random variables taking values ±1\pm 1 with probabilities 1/21 / 2. Set

Y={X, with probability rX, with probability 1rY= \begin{cases}X, & \text { with probability } r \\ X^{\prime}, & \text { with probability } 1-r\end{cases}

Find ρ(X,Y)\rho(X, Y).

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