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Linear correlation coefficient - Exercise set 1

This exercise set contains some solved exercises on the computation of the coefficient of linear correlation between two random variables. The theory needed to solve these exercises is introduced in the lecture entitled Linear correlation.

Exercise 1.1

Let X be a $2 imes 1$ discrete random vector and denote its components by X_1 and X_2. Let the support of X be: [eq1]and its joint probability mass function be:[eq2]

Compute the coefficient of linear correlation between X_1 and X_2.

nav_button Solution

The support of X_1 is:[eq3]and its marginal probability mass function is:[eq4]The expected value of X_1 is:[eq5]The expected value of $X_{1}^{2}$ is:[eq6]The variance of X_1 is:[eq7]The standard deviation of X_1 is:[eq8]The support of X_2 is:[eq9]and its marginal probability mass function is:[eq10]The expected value of X_2 is:[eq11]The expected value of $X_{2}^{2}$ is:[eq12]The variance of X_2 is:[eq13]The standard deviation of X_1 is:[eq14]Using the transformation theorem, we can compute the expected value of $X_{1}X_{2}$:[eq15]Hence, the covariance between X_1 and X_2 is:[eq16]and the coefficient of linear correlation between X_1 and X_2 is:[eq17]

Exercise 1.2

Let X be a $2 imes 1$ discrete random vector and denote its entries by X_1 and X_2. Let the support of X be: [eq18]and its joint probability mass function be:[eq19]

Compute the covariance between X_1 and X_2.

nav_button Solution

The support of X_1 is:[eq20]and its marginal probability mass function is:[eq21]The mean of X_1 is:[eq22]The expected value of $X_{1}^{2}$ is:[eq23]The variance of X_1 is:[eq24]The standard deviation of X_1 is:[eq25]The support of X_2 is:[eq26]and its probability mass function is:[eq27]The mean of X_2 is:[eq28]The expected value of $X_{2}^{2}$ is:[eq29]The variance of X_2 is:[eq30]The standard deviation of X_2 is:[eq31]The expected value of the product $X_{1}X_{2}$ can be derived using the transformation theorem:[eq32]Therefore, putting pieces together, the covariance between X_1 and X_2 is:[eq33]and the coefficient of linear correlation between X_1 and X_2 is:[eq34]

Exercise 1.3

Let [eq35] be an absolutely continuous random vector with support: [eq36]and let its joint probability density function be:[eq37]Compute the covariance between X and Y.

nav_button Solution

The support of Y is:[eq38]When $y otin R_{Y}$, the marginal probability density function of Y is 0, while, when $yin R_{Y}$, the marginal probability density function of Y can be obtained by integrating x out of the joint probability density as follows:[eq39]Thus, the marginal probability density function of Y is:[eq40]The expected value of Y is:[eq41]The expected value of $Y^{2}$ is:[eq42]The variance of Y is:[eq43]The standard deviation of Y is:[eq44]The support of X is:[eq45]When $x otin R_{X}$, the marginal probability density function of X is 0, while, when $xin R_{X}$, the marginal probability density function of X can be obtained by integrating $y$ out of the joint probability density as follows:[eq46]We do not explicitly compute the integral, but we write the marginal probability density function of X as follows:[eq47]The expected value of X is:[eq48]The expected value of $X^{2}$ is:[eq49]The variance of X is:[eq50]The standard deviation of X is:[eq51]The expected value of the product $XY$ can be computed using the transformation theorem:[eq52]Hence, using the covariance formula, the covariance between X and Y is:[eq53]and the coefficient of linear correlation between X and Y is:[eq54]

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