2.1 Distributions of Two Random Variables

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TODO: Marginal distributions

Definition 2.1.1: Random vector

Let

• $\mathcal{C}$ : Sample space
• $X_1,X_2$ : Random variables

If $X_1$ and $X_2$ assigns to each $c\in \mathcal{C}$ one and only one ordered pair of numbers $X_1(c)=x_1,X_2(c)=x_2$

Then

• We say $(X_1,X_2)$ is a random vector
• We say $\mathcal{D}=\{(x_1,x_2):x_1=X_1(c),x_2=X_2(c),c\in \mathcal{C}\}$ is the space of $(X_1,X_2)$

We often denote random vectors using $\mathbf{X}=(X_1,X_2{)}^{\prime }$

Also, we often use $(X,Y)$ to denote random vectors

Cumulative distribution function (cdf)

$F_{X,Y}(x,y)=P[\{X\le x\}\cap \{Y\le y\}],\forall (x,y)\in {\mathbb{R}}^2$

We may also write $P[\{X\le x\}\cap \{Y\le y\}]$ as $P[X\le x,Y\le y]$

Joint cumulative distribution function (cdf)

As exercise 2.1.3 shows,

$$P[a_1<X\le b_1,a_2<Y\le b_2]=F(b_1,b_2)-F(a_1,b_2)-F(b_1,a_2)+F(a_1,a_2)$$

Hence, all induced probabilities of this form can be formulated in terms of the cdf. We call this cdf the joint cumulative distribution function of $(X,Y)$

Discrete random vector

Let $(X,Y)$ : Random vector

If space of $(X,Y)$ is finite or countable

Then

• We say $(X,Y)$ is a discrete random vector
• Both $X,Y$ are discrete random variable

Joint probability mass function (pmf)

Let $(X,Y)$ : Random vector

If

$$p_{X,Y}(x,y)=P[X=x,Y=y],\forall (x,y)\in \mathcal{D}$$

Then we say $p_{X,Y}$ is the joint probability mass function (pmf) of $(X,Y)$

As with random variables, the joint pmf have the following 2 properties:

1. $0\le p_{X,Y}(x,y)\le 1$
2. ${\sum }_{(x,y)\in \mathcal{D}}{p}_{X,Y}(x,y)=1$

Also, for an event $B\in \mathcal{D}$, we have

$$P[(X,Y)\in B]=\sum _{(x,y)\in \mathcal{D}}{p}_{X,Y}(x,y)$$

Support of discrete random vector

Let $(X,Y)$ : Random vector

If ${\mathcal{S}}_{X,Y}=\{(x,y):p_{X,Y}(x,y)>0\}$

Then ${\mathcal{S}}_{X,Y}$ is the support of $(X,Y)$

Continuous random vector

Let $(X,Y)$ : Random vector

If cdf of $(X,Y)$ is continuous

Then we say $(X,Y)$ is a continuous random vector

Joint probability density function (pdf)

Let $(X,Y)$ : Random vector

If

$$F_{X,Y}(x,y)={\int }_{-\infty }^x{\int }_{-\infty }^y{f}_{X,Y}(v,w)dvdw,\forall (x,y)\in {\mathbb{R}}^2$$

Then we say $F_{X,Y}$ is the joint probability density function (pmf) of $(X,Y)$

And

$$\frac{{\partial }^2{F}_{X,Y}(x,y)}{\partial x\partial y}=f_{X,Y}(x,y)$$

As with random variables, the joint pdf have the following 2 properties:

1. $f_{X,Y}(x,y)\ge 0$
2. $\int {\int }_{\mathcal{D}}{f}_{X,Y}(x,y)dxdy=1$

Also, for an event $A\in \mathcal{D}$, we have

$$P[(X,Y)\in A]=\int {\int }_A{f}_{X,Y}(x,y)dxdy$$ Link to original

Theorem 2.1.1

Let

• $(X_1,X_2)$ : Random vector
• $Y_1=g_1(X_1,X_2)$
• $Y_2=g_2(X_1,X_2)$
• $E(Y_1),E(Y_2)$ exist

Then

$$E(k_1{Y}_1+k_2{Y}_2)=k_{1}E(Y_1)+k_{2}E(Y_2),\forall k_1,k_2\in \mathbb{R}$$

Definition 2.1.2: mgf of random vector of two random variables

Let $\mathbf{X}=(X_1,X_2{)}^{\prime }$ : Random vector

If

$$M_{X_1,X_2}(t_1,t_2)=E(e^{t_1{X}_1+t_2{X}_2})\text{ exists},|t_1|<h_1,|t_2|<h_2,h_1\in \mathbb{P},h_2\in \mathbb{P}$$

Then we say $M_{X_1,X_2}$ is the mgf of $\mathbf{X}$

Definition 2.1.3: Expected value of random vector of two random variables

Let $\mathbf{X}=(X,Y{)}^{\prime }$ : Random vector

If $E(X),E(Y)$ exist

Then $E[\mathbf{X}]=E\left[\begin{array}{c}E(X)\\ E(Y)\end{array}\right]$ exist