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Chapter 5 Bivariate Distributions

The concepts discussed in this chapter are joint and marginal distributions, independence and conditional distributions, for both discrete and continuous random variables. There are several ways through this material and our choice is to deal with discrete and continuous separately. We give a quick, but complete, run through of these distributions in the discrete case, and then follow this with a more extensive treatment of the continuous case.

In the following we will concentrate on situations where there are only two random variables $X$ and $Y$ . In subsequent chapters we discuss the extension to more than two random variables.

5.1 Introduction and Motivation

5.2 Cumulative Distribution Function

5.3 Discrete Random Variables

5.4 Continuous Random Variables

5.5 Marginal Distributions

6.5 Key definitions and Relationships

Let $(X,Y)$ be a bivariate rv and $\boldsymbol{X}=(X_{1},\dots,X_{n})^{t}$ and $\boldsymbol{Y}=(Y_{1},\dots,Y_{m})$ be vector rvs.

1.

Bivariate expectation: for a discrete rv $\operatorname{\mathsf{E}}\left[{g(X,Y)}\right]=\sum_{i=-\infty}^{\infty}\sum_{% j=-\infty}^{\infty}p_{X,Y}(i,j)g(i,j)$ . For a continuous rv $\operatorname{\mathsf{E}}\left[{g(X,Y)}\right]=\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}f_{X,Y}(s,t)g(s,t)\,\mathrm{d}s\,\mathrm{d}t$ .
2.

Linearity: $\operatorname{\mathsf{E}}\left[{ag(X,Y)+bh(X,Y)}\right]=a\operatorname{\mathsf% {E}}\left[{g(X,Y)}\right]+b\operatorname{\mathsf{E}}\left[{h(X,Y)}\right]$ .
3.

If $X$ and $Y$ are independent then $\operatorname{\mathsf{E}}\left[{g(X)h(Y)}\right]=\operatorname{\mathsf{E}}% \left[{g(X)h(Y)}\right]$ .
4.

The conditional expectation of $X$ given $Y=y~{}$ is $\operatorname{\mathsf{E}}\left[{g(X)|Y=y}\right]=\sum_{i=-\infty}^{\infty}p_{X% |Y}(i|y)g(i)$ if $X$ is a discrete rv, and $\operatorname{\mathsf{E}}\left[{g(X)|Y=y}\right]=\int_{-\infty}^{\infty}f_{X|Y% }(t|y)g(t)\,\mathrm{d}t$ if $X$ is continuous.
5.

The conditional variance of $X$ given $Y=y~{}$ is ${\operatorname{\mathsf{Var}}}\left[{X|Y=y}\right]=\operatorname{\mathsf{E}}% \left[{X^{2}|Y=y}\right]-\operatorname{\mathsf{E}}\left[{X|Y=y}\right]^{2}$ .
6.

Tower: $\operatorname{\mathsf{E}}\left[{X|Y}\right]$ is a function of $Y$ and hence a random variable; $\operatorname{\mathsf{E}}\left[{\operatorname{\mathsf{E}}\left[{X|Y}\right]}% \right]=\operatorname{\mathsf{E}}\left[{X}\right]$ .
7.

The moment generating function, $M_{X}(t)=\operatorname{\mathsf{E}}\left[{e^{tX}}\right]$ , uniquely determines the distribution of $X$ . For integer $k$ , $\operatorname{\mathsf{E}}\left[{X^{k}}\right]=M^{(k)}_{X}(0)$ .
8.

If $X$ and $Y$ are independent then $M_{X+Y}(t)=M_{X}(t)M_{Y}(t)$ .