Joint Distribution

Definition:

• PDF
• For any two random variables $X$ and $Y$, which will make up a new random variable of different distribution with distinct observations consists of both $X$ and $Y$
  • Unlike related variable, where observations of the same distribution is added
• For both discrete and continuous
• $X$ and $Y$ are jointly continous for all continuous random variables, having the property that for every set $C$ of pairs, if there exists a joint probability density function $f(x,y)$ that $P\{(X,Y)\in C\}=\int {\int }_{(x,y\in C)}f(x,y)dxdy$
  • Where $f(x,y)$ is the probability of selecting both of them
  • Defining $C=\{(x,y):x\in A,y\in B\}$
  • Then $P\{X\in A,Y\in B\}={\int }_B{\int }_{A}f(x,y)dxdy$
  • Equivalently $F(a,b)=P\{X\in (-\infty ,a],Y\in (-\infty ,b]\}={\int }_{-\infty }^b{\int }_{-\infty }^{a}f(x,y)dxdy$
  • $f(a,b)=\frac{{\partial }^2}{\partial a\partial b}F(a,b)$
• If $X$ and $Y$ are jointly continuous, they are individually continuous, and their PDF is:
  • $f_X(x)={\int }_{-\infty }^{\infty }f(x,y)dy$
    • therefore. expectation of 1 variable is $E[X]=\int x.f_X(x)dx=\int {\int }_{-\infty }^{\infty }x.f(x,y)dydx$
  • $f_Y(y)={\int }_{-\infty }^{\infty }f(x,y)dx$
• $P(X\in A)=P\{X\in A,Y\in (-\infty ,\infty )\}={\int }_A{\int }_{-\infty }^{\infty }f(x,y)dydx={\int }_A{f}_X(x)dx$
  • Double integrals over general regions

Inequality between jointly random variable:

• $P(X_1<X_2)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{X^2}f(x_1,x_2)d{x}_{1}d{x}_2$
• $P(\frac{X_1}{X_2}<Z)\to P(Z<a)=P(\frac{X_1}{X_2}<a)=P(X_1<a{X}_2)={\int }_{X_2^l}^{\infty }{\int }_{X_1^l}^{a{X}_2}f(x_1,x_2)d{x}_{1}d{x}_2$

For number of jointly random variable, $n>2$

• We can also define joint probability distributions for $n$ random variables in exactly the same manner as we did for $n=2$

Independent random variables

• If the random variable $X$ and $Y$ for any two sets of real numbers $A$ and $B$ $P\{X\in A,Y\in B\}=P\{X\in A\}.P\{Y\in B\}$
  • $P\{X\le A,Y\le B\}=P\{X\le a\}.P\{Y\le b\}$
  • $F(a,b)=F_X(a).F_Y(b)\text{for all }a,b$
• 2 continuous random variables are independent if and only if their JPDF can be expressed as: $f_{X,Y}=h(x).g(y)-\infty <x,y<\infty$

Expectation of function of joint continuous variable:

• If $X$ and $Y$ have a Joint Distribution $f(x,y)$, then $E[g(X,Y)]={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }g(x,y)f(x,y)dxdy$

Bayes theorem

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