Distribution and density functions of a linear transformation

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