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$3.2$ Distributions and probability mass functions
There are two main types of random variables used in practice: discrete r.v.s an continuous r.v.s. In this chapter and the next, our focus is on discrete r.v.s. Con tinuous r.v.s are introduced in Chapter 5 .
Definition 3.2.1 (Discrete random variable). A random variable $X$ is said to b discrete if there is a finite list of values $a_{1}, a_{2}, \ldots, a_{n}$ or an infinite list of value $a_{1}, a_{2}, \ldots$ such that $P\left(X=a_{j}\right.$ for some $\left.j\right)=1$. If $X$ is a discrete r.v., then th
Random variables and their distributions
finite or countably infinite set of values $x$ such that $P(X=x)>0$ is called th support of $X$.

Most commonly in applications, the support of a discrete r.v, is a set of integers In contrast, a continuous r.v, can take on any real value in an interval (possibl even the entire real line); such r.v.s are defined more precisely in Chapter $5 .$ is also possible to have an r.v. that is a hybrid of discrete and continuous, suc as by flipping a coin and then generating a discrete r.v. if the coin lands Head understanding such r.y.s is to understand discrete and continuous r. $y .8 .$
Given a randem variable, we weuld like to be able to describe its behavior using th language of probability. For example, we might want to answer questions about th probability that the r.v. will fall into a given range: if $L$ is the lifetime earnings a randomly chosen U.S. college graduate, what is the probability that $L$ exceeds million dollars? If $M$ is the number of major earthquakes in California in the nex five years, what is the probability that $M$ equals 0 ?

The distribution of a random variable provides the answers to these questions: specifies the probabilities of all events associated with the r.v., such as the probi bility of it equaling 3 and the probability of it being at least 110 . We will see tha there are several equivalent ways to express the distribution of an r.v. For a discret $\mathrm{r}_{\mathrm{V}}$, the most natural way to do so is with a probability mass function, which w now define,

Definition $3.2 .2$ (Probability mass function). The probability mass function (PMF) of a discrete r.v. $X$ is the function $p_{X}$ given by $p_{X}(x)=P(X=x)$. Not that this is positive if $x$ is in the support of $X$, and 0 otherwise.
w $3.2 .3 .$ In writing $P(X=x)$, we are using $X=x$ to denote an event, consisting of all outcomes $s$ to which $X$ assigns the number $x$. This event is also written a ${X=x} ;$ formally, ${X=x}$ is defined as ${s \in S: X(s)=x}$, but writing ${X=x$ is shorter and more intuitive, Going back to Example 3,1,2, if $X$ is the numbe of Heads in two fair coin tosses, then ${X=1}$ consists of the sample outcome $H T$ and $T H$, which are the two outcomes to which $X$ assigns the number 1. Sinc ${H T, T H}$ is a subset of the sample space, it is an event. So it makes sense to tal ${H T, T H}$ is a subset of the sumple space, it is an event. So it makes sense to tal about $P(X=1)$, or more generally, $P(X=x)$. If ${X=x}$ were anything othe than an event, it would make no sense to calculate its probability! It does not make sense to write ” $P(X) “$; we can only take the probability of an event, not of an r.
Let’s look at a few examples of PMFs.
Example $3.2 .4$ (Coin tosses continued). In this example we’ll find the PMFs all the random variables in Example 3.1.2, the example with two fair coin tosses Here are the r.v.s we defined, along with their PMEs:

• $X$, the number of Heads. Since $X$ equals 0 if TT occurs, 1 if HT or TH occurs
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and 2 if $H H$ occurs, the PMF of $X$ is the function $p_{X}$ given by
\begin{aligned} &p_{X}(0)=P(X=0)=1 / 4 \ &p_{X}(1)=P(X=1)=1 / 2 \ &p_{X}(2)=P(X=2)=1 / 4 \end{aligned}
and $p_{X}(x)=0$ for all other values of $x$.
$P(Y=y)=P(2-X=y)=P(X=2-y)=p_{X}(2-y)$

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$3.2$ 分布和概率质量函数