23 Lecture 22, March 04, 2024
23.1 Receipt to find the distribution of the transformed random varaible \(Y=g(X)\).
In class, we have an easy three steps receipt:
Let \(Y=g(X)\). In general, we can use the following steps to find the pdf of \(Y=g(X)\):
Using the support of \(X\), find the support of \(Y=g(X)\) denoted by \(\text{supp}(Y)=\{y\in\mathbb{R}:f_Y(y)>0\}\)
Express the cdf of \(Y=g(X)\) using the cdf of \(X\): $F_Y(y)=P(g(X)y)=$.
Compute the pdf of \(Y=g(X)\) by differentiating \(f_y(y)=F_y'(y)\)
Notes: If the function \(g\) is invertible and differentiable with inverse \(g^{-1}\) on the support of \(Y\), then \[ f_Y(y)= |(g^{-1})'(y)| f_X(g^{-1}(y)),\quad y\in\text{supp}(Y)\]
Question: When the function \(g\) is not strictly increasing (or decreasing) over the support of \(X\), then we must be careful when rewriting the inequality \(P(g(X)\leq y)\).
23.2 Quantile
Definition 23.1 (Quantile) Let \(p\in[0,1]\). The \(100\times p\)th percentile (or \(100\times p\%\) quantile) of the distribution of \(X\) with cdf \(F_X\) is the value \(c_p\) given by \[ c_p = \inf\{x\in\mathbb{R}: F_X(x) \geq p \}\]
- The infimum of a set \(A\) is the largest lower bound of \(A\) (e.g., \(\inf\{x\in\mathbb{R}: 0<x<1\}=0\))
- The quantile function \(p\mapsto c_p\) is also called generalized invere function.
- The probability that \(X\) is at most \(c_p\) is at least \(100\times p\)%. More precisely, \(c_p\) is the smallest value \(c\) so \(P(X\leq c)\) is at least \(p\). *The of a distribution is its 50% quantile.
- If the distribution function \(F_X\) is continuous and strictly increasing, it has an inverse \(F^{-1}\), and we get \[ c_p = F^{-1}(p)\]
- In the previous clicker question, we computed the 95% quantile.