Functions of Random Variables
Learning Outcomes
Functions of Random Variables
Finding PDFs using the distribution function
Finding the PDF of a function of random variables
Using Moment Generating Functions
Function of Random Variables
Function of Random Variables
Obtaining the PDFs
Using the Distribution Function
Let there be a random variable \(X\) with a known distribution function \(F_X(x)\), the density function for the random variable \(Y=g(X)\) can be found with the following steps
- Find the region of \(Y\) in the space of \(X\), find \(g^{-1}(y)\)
- Find the region of \(Y\le y\)
- Find \(F_Y(y)=P(Y\le y)\) using the probability density function of \(X\) over region \(Y\le y\)
- Find \(f_Y(y)\) by differentiating \(F_Y(y)\)
Example 1
Let \(X\) have the following probability density function:
\[ f_X(x)=\left\{\begin{array}{cc} 2x & 0\le x \le 1 \\ 0 & \mathrm{otherwise} \end{array} \right. \]
Find the probability density function of \(Y=3X-1\)?
Using the PDF
Let there be a random variable \(X\) with a known distribution function \(F_X(x)\), if the random variable \(Y=g(X)\) is either increasing or decreasing, than the probability density function can be found as
\[ f_Y(y) = f_X\{g^{-1}(y)\}\left|\frac{dg^{-1}(y)}{dy}\right| \]
Example 2
Let \(X\) have the following probability density function:
\[ f_X(x)=\left\{\begin{array}{cc} \frac{3}{2}x^2 + x & 0\le y \le 1 \\ 0 & \mathrm{otherwise} \end{array} \right. \]
Find the probability density function of \(Y=5-(X/2)\)?
Using the MGF
Using the uniqueness property of Moment Generating Functions, for a random variable \(X\) with a known distribution function \(F_X(x)\) and random variable \(Y=g(X)\), the distribution of \(Y\) can be found by:
- Find the moment generating function of \(Y\), \(M_Y(t)\).
- Compare \(M_Y(t)\), with known moment generating functions. If \(M_Y(t)=M_V(t)\), for all values \(t\), them \(Y\) and \(V\) have identical distributions.
Example 3
Let \(X\) follow a normal distribution with mean \(\mu\) and variance \(\sigma^2\). Find the distribution of \(Z=\frac{X-\mu}{\sigma}\).
Example 4
Let \(Z\) follow a standard normal distribution with mean \(0\) and variance \(1\). Find the distribution of \(Y=Z^2\)
