Continuous Random Variables

Week 7

Maghfira Ramadhani

Oct 01, 2025

Plan

In today’s class we will learn In today’s class we will:

1. Define continuous random variables
2. Use probability density function (p.d.f) and cumulative distribution function (c.d.f.) to describe continuous r.v. distribution
3. Use p.d.f. to compute population descriptive statistics: population quantiles, mean, variance, standard deviation
4. Use joint (and conditional) p.d.f. and joint (and conditional) c.d.f. to:
   1. describe multiple continuous r.v. distribution: population covariance and correlation
   2. check independence of two continuous r.v.

Textbook Reference: JA 10; SDG 3.2

Continuous random variables

A continuous random variable is a random variable that can take any value on some interval or intervals of the real line, including perhaps the entire real line, and for which the probability of any specific outcome \(x^*\) occuring equals zero.

Example:

1. Weekly earning: \(S=[0,\infty)\)

2. Unemployment rate: \(S=[0,1]\)

Comparison to discrete r.v.

	Discrete	Continuous
Sample space	\[ S=\{x_1^*,x_2^*,\ldots x_n^*\} \]	\[ S=\{X|X\subset \mathbb{R}\} \]
Definition	\[ p_X(x_k^*)=P(X=x_k^*)\quad \text{ (pmf/pf)} \]	\[ f_X(x)=F'(x)\quad \text{ (pdf)} \]
Computing probability	\[ P(X\in A)=\sum_{x_k^*\in A}p_X(x_k^*) \]	\[ P(X\in A)=\int_{A}f(x)dx \]
CDF	\[ F_X(x_k^*)=P(X\leq x_k^*)=\sum_{y\leq x_k^*}p_X(y) \]	\[ F_X(x)=P(X\leq x)=\int_{-\infty}^x f(y)dy \]

Note that the total probability over the sample space should add up to 1.

Descriptive statistics

	Discrete	Continuous
Expected value or mean	\[ \mu_X=E(X)=\sum_{k} x_k^* p_X(x_k^*). \]	\[ \mu_X=E(X)=\int_{-\infty}^\infty xf(x)dx. \]
Variance	\[ \sigma_X^2=Var(X)=\sum_{k} (x_k^*-\mu_X)^2 p_X(x_k^*). \]	\[ \sigma_X^2=Var(X)=\int_{-\infty}^\infty (x-\mu_X)^2 f(x)dx. \]
Standard Deviation	\[ \sigma_X=\sqrt{Var(X)} \]	\[ \sigma_X=\sqrt{Var(X)} \]

Population quantiles

Recall when we define sample quantiles in Week 3. Here we define the population analog:

For any \(q\), with \(0<q<1\), the population quantile \(\tau_{X,q}\) of a continuous random variable \(X\) is the value for which

\[P(X\leq \tau_{X,q})=F_X(\tau_{X,q})=q.\]

Mathematically, \(\tau_{X,q}=F^{-1}(q)\) is the inverse of \(F(x)\).

Applied Exercise 1

A random variable \(X\sim\text{Uniform}(0,2)\). Compute the population mean, standard deviation, the quantile function, and the median (quantiles at \(q=0.5\)).

Applied Exercise 1

Applied Exercise 1

Applied Exercise 1

Applied Exercise 2

X is a continuous random variable with following pdf:

\[f_X(x)=\begin{cases}ax,&-1\leq x<0\\ x,& 0\leq x<1\\ 0,&\text{otherwise}\end{cases}\]

Determine the value of \(a\), the cdf \(F_X(x)\), and the quantile function \(\tau_{X,q}\), and the population mean \(E(X)\).

Applied Exercise 2

Applied Exercise 2

Applied Exercise 2

Multiple continuous random variables

	Definition	Properties
Joint pdf	\[ f_{XY}(x,y) \] \[f_{XY}(x,y)\geq0, \quad\forall(x,y) \] \[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{XY}(x,y)dx dy=1, \quad\forall(x,y) \]	\[\text{Marginal pdf of X: }f_X(x)=\int_{-\infty}^{\infty}f_{XY}(x,y)dy, \quad\forall(x,y) \] \[\text{Marginal pdf of Y: }f_Y(y)=\int_{-\infty}^{\infty}f_{XY}(x,y)dx, \quad\forall(x,y) \]
Joint cdf	\[ F_{XY}(x_0,y_0)=\int_{-\infty}^{x_0}\int_{-\infty}^{y_0}f_{XY}(x,y)dx dy \]
Conditional pdf	\[ f_{X|Y}=f_{XY}(x,y)/f_Y(y), \quad \forall(x,y),f_Y(y)>0 \]	\[ f_{Y|X}=f_{XY}(x,y)/f_X(x), \quad \forall(x,y),f_X(x)>0 \]
Conditional cdf	\[ F_{X|Y}(x|y)=P(X\leq x|Y=y)\]	\[=\int_{-\infty}^xf_{XY}(x,y)/f_Y(y), \quad \forall(x,y),f_Y(y)>0 \]

Population covariance and correlation

The population covariance of continuous random variables \(X\) and \(Y\) is

\[\sigma_{XY}=Cov(X,Y)=E[(X-\mu_X)(Y-\mu_Y)]=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x-\mu_X)(y-\mu_y)f_{XY}dxdy.\]

The population correlation is then \[\rho_{XY}=Corr(X,Y)=\frac{\sigma_{XY}}{\sigma_X \sigma_Y}\]

Independence of random variables

The random variables \(X\) and \(Y\) are independent if and only if

\[F_{XY}(x,y)=F_X(x)F_Y(y)\] for every possible pair \((x,y)\).

Applied Exercise 3

Random variables \(X\) and \(Y\) has the following joint pdf:

\[f_{XY}(x,y)=\begin{cases}axy,&0\leq x<y\leq 1\\ 0,&\text{otherwise}\end{cases}\]

Determine the value of \(a\), the probability \(P(X\leq 2Y)\), the marginal probability \(f_X(x)\), the expected value \(E(X)\), the variance \(Var(X)\), and the conditional expected value \(E(Y|X=0.5)\). Also, check if \(X\) and \(Y\) are independent random variables.

Applied Exercise 3

Applied Exercise 3

Applied Exercise 3

Applied Exercise 3

End of Lecture