5.3 - continuous marginal and conditional distributions

Marginal CDFs

Recall that if \(X\) and \(Y\) have joint CDF given by \(F(x,y)\), then the marginal CDFs are defined as:

\[F_X(x) = P(X\leq x) =\lim_{y\rightarrow \infty} F(x,y)\] \[F_Y(y) = P(Y\leq y) =\lim_{x\rightarrow \infty} F(x,y)\]

This result holds for any pair of random variables (jointly discrete, jointly continuous, mixture).

Marginals for jointly continuous random variables

If \(X\) and \(Y\) are jointly continuous with joint pdf \(f(x,y)\), then the marginal probability density functions are defined as:

\[f_X(x) = \int_{-\infty}^\infty f(x,y) dy\]

\[f_Y(y) = \int_{-\infty}^\infty f(x,y) dx\]

Another way of finding the marginal pdfs is to differentiate the marginal CDFs:

\[f_X(x) = \frac{d}{dx} F_X(x)\] \[f_Y(y) = \frac{d}{dy} F_Y(y)\]

Note on notation

Let’s clarify this subscript/argument notation!
- \(F_X(x) = P(X\leq x)\): the marginal CDF of \(X\) evaluated at \(x\) (analogously defined for \(F_Y(y)\))
- \(f_Y(y)\): the marginal pdf of \(Y\) evaluated at \(y\) (analogously defined for \(f_X(x)\))
The subscripts define the function, not the argument! (Think: \(f_Y(\cdot)\) and \(f_X(\cdot)\) as \(h(\cdot)\) and \(g(\cdot)\))
Example: suppose \(Y\sim EXP(\lambda)\) and \(X\sim UNIF(0,1)\).

\[F_Y(x) = \begin{cases} 0 & x \leq 0 \\ 1-e^{-\lambda x}& x> 0 \end{cases}\]

\[f_Y(x) = \begin{cases}\lambda e^{-\lambda x} & x> 0\\ 0 & otherwise \end{cases}\]

\[F_X(y) =\begin{cases} 0 & y < 0 \\ y &0 \leq y \leq 1 \\ 1 & y > 1\end{cases}\]

\[f_X(y) = \begin{cases} 1 & 0 \leq y \leq 1 \\ 0 & otherwise \end{cases}\]

Example: wedge over \((0,1)^2\)

Suppose \(X\) and \(Y\) are jointly continuous with joint pdf:

\[f(x,y) = \begin{cases} 2x & 0 < x < 1, 0 < y < 1 \\ 0 & otherwise \end{cases}\]

Support:

Desmos screenshot (interactive: https://www.desmos.com/3d/rrimrnv8mt)

Views of joint density along X and Y axes

Finding the marginals

\[f_X(x) = \int_0^1 2x\ dy = 2xy\big|_{y=0}^1 = \begin{cases} 2x& 0 < x < 1\\ 0 & otherwise \end{cases}\] \(X \sim BETA(2,1)\)

\[f_Y(y) = \int_0^1 2x\ dx = x^2\big|_{x=0}^1 = \begin{cases} 1 & 0 < y < 1\\ 0 & otherwise \end{cases}\] \(Y \sim UNIF(0,1)\)

Conditional densities

If \(X\) and \(Y\) are jointly continuous random variables with joint pdf \(f(x,y)\), then the conditional density functions are defined accordingly:

For any \(y\) such that \(f_Y(y)>0\):

\[f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}\]

For any \(x\) such that \(f_X(x)>0\):

\[f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)}\]

Conditionals have all the properties of a univariate pdf:
- \(f_{X|Y}(x|y)\geq 0\ \forall\ x\)
- \(\int_{-\infty}^\infty f_{X|Y}(x|y)\ dx = 1\)
- \(f_{Y|X}(y|x)\geq 0\ \forall\ y\)
- \(\int_{-\infty}^\infty f_{Y|X}(y|x)\ dy = 1\)
For ease of notation, we may at times drop the subscripts on these conditionals.

Conditional CDFs

If \(X\) and \(Y\) are jointly continuous random variables with conditional pdfs as defined previously, then:

\[F_{X|Y}(x|y) = P(X \leq x|Y=y) = \int_{-\infty}^x f_{X|Y}(t|y)\ dt\] \[F_{Y|X}(y|x) = P(Y \leq y|X=x) = \int_{-\infty}^y f_{Y|X}(t|x)\ dt\]

As in univariate case, we can differentiate conditional CDFs to yield conditional pdfs:

\[f_{X|Y}(x|y) = \frac{d}{dx} F_{X|Y}(x|y)\] \[f_{Y|X}(y|x) = \frac{d}{dy} F_{Y|X}(y|x)\]

Example: uniform over triangle

Suppose \(X\) and \(Y\) are jointly continuous with joint pdf:

\[f(x,y) = \begin{cases} \frac{1}{2} & 0 < x < y < 2 \\ 0 & otherwise \end{cases}\]

Find \(P(X < 1/2 | Y = 1.5)\).

Process:

Find marginal pdf of \(Y\),
Find conditional pdf \(f_{X|Y}\),
Integrate to find conditional CDF \(F_{X|Y}\).

Example: uniform over triangle

Support with horizontal strip:

\[f_Y(y) = \int_0^y \frac{1}{2}\ dx =\frac{x}{2}\big|_{x=0}^y= \begin{cases} y/2 & 0 < y < 2 \\ 0 & otherwise \end{cases}\]

For \(0 < y < 2\): \[f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)} = \frac{1/2}{y/2} = \begin{cases} \frac{1}{y} & 0 < x < y \\ 0 & otherwise \end{cases}\] Note: \(X|Y=y \sim UNIF(0,y)\)!

For \(0 < x < y\): \[P(X \leq x | Y=y) = \int_0^x f_{X|Y}(t|y) dt= \int_0^x 1/y\ dt = x/y\] \[F_{X|Y}(x|y) = \begin{cases} 0 & x \leq 0 \\ x/y & 0 < x < y \\ 1 & x \geq y \end{cases}\]

\[\therefore P(X<1/2 | Y=1.5) = F_{X|Y}(0.5|1.5) = \frac{1}{3}\]

Example: mouse ears

Let \(X\) and \(Y\) have the following joint pdf:

\[\scriptsize f(x,y) = \left\{\begin{array}{ll} 30xy^2 & x-1\leq y \leq 1-x;\ 0\leq x \leq 1 \\ 0 & otherwise\\ \end{array}\right.\]

(Fully interactive: https://www.desmos.com/3d/zhoozavkw8)

Tasks:

Find marginal pdfs
Find \(P(Y>0|X=0.75)\)
- Find \(f_{Y|X}\)
- Find \(F_{Y|X}\)

Marginal of \(X\)

Support: \(x-1\leq y \leq 1-x, 0 \leq x \leq 1\)

\[f_X(x) = \int_{x-1}^{1-x} 30xy^2\ dy\]

\[=10xy^3 \big|_{y=x-1}^{1-x} = 10x(1-x)^3-10x(x-1)^3\] \[= 10x(1-x)^3-10x(-1)^3(1-x)^3\] \[= \begin{cases} 20x(1-x)^3 & 0 < x < 1 \\ 0 & otherwise \end{cases}\]

\[X\sim BETA(2,4)!\]

Marginal of \(Y\)

Support: \(x-1\leq y \leq 1-x, 0 \leq x \leq 1\)

For \(0 < y < 1\): \[f_Y(y) = \int_{0}^{1-y} 30xy^2\ dx\] \[ = 15x^2y^2\big|_{x=0}^{1-y} = 15(1-y)^2y^2\]

For \(-1 < y < 0\): \[f_Y(y) = \int_{0}^{1+y} 30xy^2\ dx\] \[ = 15x^2y^2\big|_{x=0}^{1+y} = 15(1+y)^2y^2\]

\[f_Y(y) = \begin{cases}15y^2(1-|y|)^2& -1 < y < 1 \\ 0 & otherwise \end{cases}\]

Conditional pdf \(f_{Y|X}\)

For \(0 < x < 1\):

\[f_{Y|X}(y|x) = \frac{f(x,y)}{f_{X}(x)} = \frac{30xy^2}{20x(1-x)^3}\]

\[= \begin{cases} \frac{3y^2}{2(1-x)^3}& x-1 \leq y \leq 1-x \\ 0 & otherwise \end{cases} \]

Conditional CDF \(F_{Y|X}\)

For \(x-1 < y < 1-x\):

\[F_{Y|X}(y|x) = \int_{x-1}^y\frac{3t^2}{2(1-x)^3}\ dt = \frac{1}{2(1-x)^3}t^3\big|_{t=x-1}^y = \frac{y^3-(x-1)^3}{2(1-x)^3}\]

\[F_{Y|X}(y|x) = \begin{cases} 0 & y \leq x-1 \\ \frac{y^3-(x-1)^3}{2(1-x)^3} & x-1 < y < 1-x \\ 1 & y \geq 1-x \end{cases}\]

\(P(Y>0|X=0.75)\)

\[F_{Y|X}(y|0.75) = \begin{cases} 0 & y \leq -0.25 \\ \frac{y^3-(-0.25)^3}{2(0.25)^3} & -0.25 < y < 0.25 \\ 1 & y \geq 0.25 \end{cases}\]

\[\therefore P(Y>0|X=0.75) = 1-F_{Y|X}(0|0.75)= 1-\frac{0^3-(-0.25)^3}{2(0.25)^3} = 0.5\]