24
Mar 17

Distribution and density functions of a linear transformation

category: Dougherty Introduction to Econometrics, EC2020, EC2020 Elements of econometrics, Econometrics, The College Board 1 Comment

Distribution and density functions of a linear transformation - two short derivations to read during breakfast.

Distribution function of a linear transformation

Let X be a random variable and let Y=aX+b be its linear transformation (here a,b are some real numbers and a\ne0, otherwise Y is not random). If the distribution function F_X is known, what will be the distribution function of Y?

The answer is obtained in one line if you know the definition of the distribution function:

(1) F_Y(y)=P(Y\le y)=P(aX+b\le y)=P(X\le\frac{y-b}{a})=F_X(\frac{y-b}{a}).

For the inequalities aX+b\le y and X\le\frac{y-b}{a} to be equivalent, we have to assume that a>0 (for applications this is enough). The case a<0 is left as an exercise.

Density function of a linear transformation

As above, Y is a linear transformation of X. Suppose they have densities p_X,p_Y. What is the relationship between the densities?

Recall formula (1) that links distribution and density functions. Equation (1) in terms of densities becomes

\int_{-\infty}^yp_Y(t)dt=\int_{-\infty}^{\frac{y-b}{a}}p_X(t)dt.

Let's differentiate this equation. The Newton-Leibnitz formula applied to the integral on the left gives p_Y evaluated at y. On the right, additionally, we have to use the chain rule. The result is

(2) p_Y(y)=p_X(\frac{y-b}{a})\frac{d}{dy}\frac{y-b}{a}=\frac{1}{a}p_X(\frac{y-b}{a}).

Equation (2) will be used to derive ML estimators for the linear model.

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