1.7 Continuous Random Variables

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Definition 1.7.1: Continuous random variable

Definition

Let

• $X$ : Random variable
• $F(x)$ : cdf of $X$

If $F(x)$ is continuous for all $x\in \mathbb{R}$

Then we say $X$ is a continuous random variable

Definition: Probability density function (pdf)

Definition

Let $X$ : Continuous random variable

If $ f_{X}(x) = P[X=x] = 0, \quad \forall x\in \mathbb{R} $$

Then we say that $f_X(x)$ is the probability density function (pdf) of $X$

Definition: Support of continuous random variable

Definition

Let

• $X$ : Continuous random variable
• $\mathcal{D}$ : Space of $X$
• $f_X(x)$ : pdf of $X$

Then the support of $X$ is defined as $ \mathcal{S}X = { x \in \mathcal{D} : f{X}(x)>0 } $$

Definition 1.7.2: Quantile

Definition

Let

• $0<p<1$
• $X$ : Random variable

If

• ${\xi }_p$ such that
  • $P(X<{\xi }_p)\le p$
  • $P(X\le {\xi }_p)\ge p$

Then

• We say ${\xi }_p$ is the quantile of order $p$ of $X$
• We say ${\xi }_p$ is the $(100p)$th percentile of $X$

Theorem 1.7.1: Finding the pdf of a transformation

Theorem

Let

• $X$ : Continuous random variable, with
  • pdf $f_X(x)$
  • Support ${\mathcal{S}}_X$
• $g:{\mathcal{S}}_X\to \mathbb{R}$, one-to-one and differentiable
• $Y=g(X)$, with
  • Support ${\mathcal{S}}_Y=\{y=g(x):s\in S_X\}$
• $x=g^{-1}(y)$ : Inverse of $g$
• $\frac{d}{dy}x=\frac{d}{dy}{g}^{-1}(y)$

Then pdf of $Y$ is given by $ f_{Y}(y)=f_{X}(g^{-1}(y)) \left| \frac{d}{dy}x \right|, \quad \forall y\in \mathcal{S}_{Y} $$

Exercise

Example 1.7.6

Let $X$ have the pdf

$$f(x)=\{\begin{array}{ll}4x^3& 0<x<1\\ 0& \text{elsewhere}\end{array}$$

Consider the random variable $Y=-\log X$. Here are the steps of Theorem 1.7.1 Finding the pdf of a transformation:

1. The support of $Y=-\log X$ is $(0,\infty )$
2. If $y=-\log x$, then $x=e^{-y}$
3. $\frac{d}{dy}x=-e^{-y}$
4. Thus the pdf of $Y$ is:

$$\begin{array}{rl}{f}_Y(y)& =f_X(e^{-y})|-e^{-y}|\\ & =4(e^{-y}{)}^3{e}^{-y}\\ & =4e^{-4y}\end{array}$$