Density and Distribution Function of a Dependent Random Variable

Topics: Random Variable - Probability

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Integrating and Deriving §

(theorem)

Let $X$ be an absolutely continuous random variable with density function $f_X(x)$. Let $\phi :X\to \mathbb{R}$ be a function.

Let $Y=\phi (X)$ be another random variable with distribution function $F_Y(y)=\mathbb{P}(Y\le y)$.

We can obtain $F_Y(y)$, $Y$’s distribution function, by integrating over the region on which $Y\le y$. We can then obtain $f_y(y)$, $Y$’s density function, by deriving this distribution function.

Example

Let $Y=3X-1$ and let $f_X(x)$ be given by:

$$f_X(x)=\{\begin{array}{ll}2x& \text{if }0\le x\le 1\\ 0& \text{otherwise}\end{array}$$

We’ll find $F_Y(y)$ by integrating:

$$\begin{array}{rl}{F}_Y(y)& =\mathbb{P}(Y\le y)\\ & =\mathbb{P}(3x-1\le y)\\ & =\mathbb{P}(3x\le y+1)\\ & =\mathbb{P}\left(x\le \frac{y+1}{3}\right)\\ & ={\int }_0^{\frac{y+1}{3}}{f}_X(x)dx\\ & ={\int }_0^{\frac{y+1}{3}}2xdx\\ & =\frac{(y+1{)}^2}{9}\end{array}$$

Now, notice that $f_X(x)$ is $2x$ (what we integrated) when $0\le x\le 1$. We’ll find the corresponding values for this interval, in terms of $y$ by using $x=\frac{y+1}{3}$:

• $\frac{y+1}{3}=0⟹y=-1⟹F_Y(y)=0$ when $y<-1$
• $\frac{y+1}{3}=1⟹y=2⟹F_Y(y)=1$ when $y>2$

Thus:

$$F_Y(y)=\{\begin{array}{ll}0& \text{if }y<-1\\ \frac{(y+1{)}^2}{9}& \text{if }-1\le y\le 2\\ 1& \text{if }y>2\end{array}$$

Now, we can simply derivate to find the density function:

$$f_Y(y)=\{\begin{array}{ll}\frac{2}{9}(y+1)& \text{if }-1\le y\le 2\\ 0& \text{otherwise}\end{array}$$

Variable Change Theorem §

(theorem)

Let $X$ be an absolutely continuous random variable with values in the interval $(a,b)\in \mathbb{R}$, and density function $f_X(x)$.

Let $\phi (a,b)\to \mathbb{R}$ be a strictly increasing or decreasing continuous function with a differentiable inverse.

Then, the random variable $Y=\phi (X)$ takes values in the interval $\phi (a,b)$ and its density function is:

$$f_Y(y)=\{\begin{array}{ll}{f}_X\left({\phi }^{-1}(y)\right)\left|\frac{d\left({\phi }^{-1}(y)\right)}{dy}\right|& \text{if }y\in \phi (a,b)\\ 0& \text{otherwise}\end{array}$$