Question

Let X, Y be discrete random variables with mean ux, My respectively. Define the function K(m) = f (x –HY) – m[X – ux)] (a) Show that K' (m) = -2po xy +2mo- where p is the correlation coefficient of X, Y and os, o are the variances of X, Y. (b) Find the value of m that minimizes K(m). You may assume that ox >0. Remark: The line y = mx +(uy-mux) is known as the least squares regression line.

Answer 1

The question involves demonstrating how the derivative of the function K(m), which relates to two discrete random variables X and Y, involves their variances and the correlation coefficient; it also finds the value of m that minimizes this function, corresponding to the slope of the least squares regression line.

Step-by-step explanation:

The question asks to demonstrate a relationship with the function K(m) that involves discrete random variables X and Y, their means μX and μY, their variances σ2X and σ2Y, and the correlation coefficient ρ. The function K(m) is differentiated with respect to m to find K'(m), demonstrating the relationship with the correlation coefficient and variances. Part b asks for the value of m that minimizes K(m), which corresponds to the slope of the least squares regression line for the given data. This line minimizes the sum of the squared differences between the observed values and the values predicted by the line, hence the name least squares regression line. The question is centered around the use of linear regression to analyze the relationship between two variables.