Dadidadida: Probability density function

Friday, April 11, 2014

Probability density function

Joint pdf

$\begin{aligned} \int \int f_{X,Y}(x,y) \, dx \, dy &= 1 \\ F_{X,Y}(x,y) &= P(X \le x, Y \le y) = \int_{-\infty}^y \int_{-\infty}^x f_{X,Y}(s,t) \,ds\, dt & \text{cumulative density function (CDF) } \\ f_{X,Y}(x,y) &= \frac{d^2}{dx\, dy} F_{X,Y}(x,y) \\ \end{aligned}$

From joint to marginal

$\begin{aligned} f_X(x) &= \int f_{X,Y}(x,y)\,dy \\ F_X(x) &= P(X \le x) = \int_{-\infty}^x \bigg( \int_{-\infty}^\infty f_{X,Y}(s,t) \, dt \bigg) ds \\ \end{aligned}$

$Y = aX + b$

$\begin{aligned} p_Y(y) &= \mathbb{P}(Y=y) = \mathbb{P}(y=aX+b) = \mathbb{P}(X=\frac{y-b}{a}) = p_X(\frac{y-b}{a}) & \text{discrete r.v.} \\ F_Y(y) &= \mathbb{P}(Y \le y) = \mathbb{P}(aX+b \le y) = \mathbb{P}(X \le \frac{y-b}{a}) = F_X(\frac{y-b}{a}) \\ \color{orange}{f_Y(y)} &= \frac{d}{dx}F_X(\frac{y-b}{a}) = \color{orange}{\frac{1}{|a|}\cdot f_X(\frac{y-b}{a})} & \text{continuous r.v.} \\ \end{aligned}$

$\color{red}{Z = X + Y}$

$\quad(X,Y$ independent)

$\begin{aligned} \mathbb{P}_Z(z) &= \sum_x \mathbb{P}(X=x, Y=z-x) = p_Z(z) \\ p_Z(z) &= \sum_x p_X(x) \cdot p_Y(z - x) \\ \end{aligned}$

Discrete convolution mechanics

Given $z$ and $Z=X+Y$ , find $p_Z(z)$ from $p_X(x)$ and $p_Y(y)$ .

$\begin{aligned} \color{blue}{f_{Z|X}(z|x)} &= f_{Y+X|X}(z|x) = f_{Y+X}(z) & \text{by independence of } X,Y \\ &= \color{blue}{f_Y(z-x)} & \color{orange}{f_{X+b}(x) = f_X(x-b)}, \text{ see Lec 11} \end{aligned}$

Joint pdf of Z and X

$\begin{aligned} f_{X,Z}(x,z) &= f_X(x)\cdot \color{blue}{f_{Z|X}(z|x)} = f_X(x) \cdot \color{blue}{f_Y(z-x)} \\ f_{Z}(z) &= \int_{-\infty}^\infty f_{X,Z}(x,z)\,dx = \int_{-\infty}^\infty f_X(x) \cdot \color{blue}{f_Y(z-x)} \,dx \\ \end{aligned}$

$Y = g(X)$

$\quad$ How to find $f_Y(y)$ in general ?

$\begin{aligned} F_Y(y) &= \mathbb{P}(g(X) \le y) \\ f_Y(y) &= \frac{d}{dy}F_Y(y) \\ \end{aligned}$

$Y = g(X)$ when $g$ is monotonic

$\begin{aligned} F_Y(y) &= \mathbb{P}(g(X) \le y) = \mathbb{P}(X \le h(y)) = F_X(h(y)) \\ f_Y(y) &= \frac{d}{dy}F_Y(y) = \frac{d}{dy}F_X\left(h(y)\right) = f_X\left(h(y)\right) \big\lvert \frac{d}{dy} h(y) \big\rvert \\ \end{aligned}$

Source: MITx 6.041x, Lecture 9, 11, 12.

# posted by rot13(Unafba Pune) @ 4:56 PM

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Friday, April 11, 2014

Probability density function

Joint pdf

From joint to marginal

$Y = aX + b$

$\color{red}{Z = X + Y}$

Discrete convolution mechanics

Joint pdf of Z and X

$Y = g(X)$

$Y = g(X)$ when $g$ is monotonic

Dadidadida ...

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Friday, April 11, 2014

Probability density function

Joint pdf

From joint to marginal

Y=aX+bY = aX + b

Z=X+Y\color{red}{Z = X + Y}

Discrete convolution mechanics

Joint pdf of Z and X

Y=g(X)Y = g(X)

Y=g(X)Y = g(X) when gg is monotonic

Dadidadida ...

Links

Previous Posts

Archives

$Y = aX + b$

$\color{red}{Z = X + Y}$

$Y = g(X)$

$Y = g(X)$ when $g$ is monotonic