# 1986 AnIntroToMathStats

- (Larsen & Marx, 1986) ⇒ Richard J. Larsen, and Morris L. Marx. (1986). “An Introduction to Mathematical Statistics and Its Applications, 2nd edition." Prentice Hall. ISBN:013487174X

**Subject Headings:** Statistics Textbook.

## Notes

- Includes definitions for:

## Quotes

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…

### 2.2 The Sample Space

By an *experiment* we will mean any procedure that (1) can be repeated, theoretically, an infinite number of times; and (2) has a well-defined set of possible outcomes. Thus, rolling a pair of dice qualifies as an experiment; so does measuring a hypertensive's blood pressure or doing a stereographic analysis to determine the carbon content of moon rocks. Each of the potential eventualities of an experiment is referred to as a *sample outcome*, [math]\displaystyle{ s }[/math], and their totality is called the *sample space*, *S*. To signify the member of [math]\displaystyle{ s }[/math] in [math]\displaystyle{ S }[/math], we write [math]\displaystyle{ s }[/math] In *S*. Any designated collection of sample outcomes, including individual outcomes, the entire sample space, and the null set, constitutes an *event*. The latter is said to *occur* if the outcome of the experiment is one of the members of that event.

### 2.3 The Probability Function

Consider a sample space, [math]\displaystyle{ S }[/math], and any event, [math]\displaystyle{ A }[/math], defined on [math]\displaystyle{ S }[/math]. If our experiment were performed *one* time, either [math]\displaystyle{ A }[/math] or [math]\displaystyle{ A^C }[/math] would be the outcome. If it were performed [math]\displaystyle{ n }[/math] times, the resulting set of sample outcomes would be members of [math]\displaystyle{ A }[/math] on [math]\displaystyle{ m }[/math] occasions, [math]\displaystyle{ m }[/math] being some integer between [math]\displaystyle{ 1 }[/math] and [math]\displaystyle{ n }[/math], inclusive. Hypothetically, we could continue this process an infinite number of times. As [math]\displaystyle{ n }[/math] gets large, the ratio *m/n* will fluctuate less and less (we will make that statement more precise a little later). The number that m/n convert to is called the *empirical probability* of [math]\displaystyle{ A }[/math] : that is, [math]\displaystyle{ P(A) = lim_{n → ∞}(m/n) }[/math]. … the very act of repeating an experiment under identical conditions an infinite number of times is physically impossible. And left unanswered is the question of how large [math]\displaystyle{ n }[/math] must be to give a good approximation for [math]\displaystyle{ lim_{n → ∞}(m/n) }[/math].

The next attempt at defining probability was entirely a product of the twentieth century. Modern mathematicians have shown a keen interest in developing subjects axiomatically. It was to be expected, then, that probability would come under such scrutiny … The major breakthrough on this front came in 1933 when Andrei Kolmogorov published *Grundbegriffe der Wahscheinlichkeitsrechnung* (*Foundations of the Theory of Probability.*). Kolmogorov's work was a masterpiece of mathematical elegance - it reduced the behavior of the probability function to a set of just three or four simple postulates, three if the same space is limited to a finite number of outcomes and four if [math]\displaystyle{ S }[/math] is infinite.

### 3 Random Variables

…

### 3.1 Introduction

Throughout most of Chapter 3, probability functions were defined in terms of the elementary outcomes making up an experiment's sample space. Thus, if two fair dice were tossed, a [math]\displaystyle{ P }[/math] value was assigned to each of the 36 possible pairs of upturned faces: … 1/36 … We have already seen, though, that in certain situations some attribute of an outcome may hold more interest for the experimenter than the outcome itself. A craps player, for example, may be concerned only that he throws a 7... In this chapter we investigate the consequences of redefining an experiment's sample space. … The original sample space contains 36 outcomes, all equally likely*. The revised sample space contains 11 outcomes, but the latter are *not* equally likely.*

In general, rules for redefining samples spaces - like going from (*x, y'*s to (*x* + *y*)'s are called *random variables*. As a conceptual framework, random variables are of fundamental importance: they provide a single rubric under which *all* probability problems may be brought. Even in cases where the original sample space needs no redefinition - that is, where the measurement recorded is the measurement of interests - the concept still applies: we simply take the random variable to be the identify mapping.

### 3.2 Densities and Distributions

**Definition 3.2.1.** A real-valued function whose domain is the sample space S is called a *random variable*. We denote random variables by uppercase letters, often [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], or [math]\displaystyle{ Z }[/math].

If the range of the mapping contains either a finite or a countably infinite number of values, the random variable is said to be *discrete* ; if the range includes an interval of real numbers, bounded or unbounded, the random variable is said to be *continuous*.

Associated with each discrete random variable [math]\displaystyle{ Y }[/math] is a *probability density function* (or *pdf*) [math]\displaystyle{ f(y) }[/math]. By definition, [math]\displaystyle{ f(y) }[/math] is the sum of all the probabilities associated with outcomes in [math]\displaystyle{ S }[/math] that get mapped into [math]\displaystyle{ y }[/math] by the random variable [math]\displaystyle{ Y }[/math]. That is, [math]\displaystyle{ f(y) = P(\{s\in S \vert Y(s)=y\}) }[/math]

Conceptually, [math]\displaystyle{ f_Y(y) }[/math] describes the probability structure induced on the real line by the random variable [math]\displaystyle{ Y }[/math].

For notational simplicity, we will delete all references to [math]\displaystyle{ s }[/math] and [math]\displaystyle{ S }[/math] and write:

- [math]\displaystyle{ f_Y(y) = P(Y(s)=y). }[/math]

In other words, [math]\displaystyle{ f\lt sub\gt Y\lt /sub\gt (y) }[/math] is the “probability that the random variable [math]\displaystyle{ Y }[/math] takes on the value [math]\displaystyle{ y }[/math].”

Associated with each continuous random variable [math]\displaystyle{ Y }[/math] is also a probability density function, *f _{Y}*(

*y*), but

*f*(

_{Y}*y*) in this case is

*not*the probability that the random variable [math]\displaystyle{ Y }[/math] takes on the value

*y*. Rather,

*f*(

_{Y}*y*) is a continuous curve having the property that for all [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math],

*P*(*a*≤ [math]\displaystyle{ Y }[/math] ≤*b*) =*P*({*s*(∈)*S*\vert [math]\displaystyle{ a }[/math] ≤*Y*(*s*) ≤*b*}) = Integral(a,b). “f_{Y}*(*y*)*dy*]*

### 3.3 Joint Densities

Section 3.2 introduced the basic terminology for describing the probabilistic behavior of a *single* random variable …

- Definition 3.3.1.

(a) Suppose that [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are two discrete random variables defined ont he same sample space *S*. The *[[joint probability density function of X and Y* (or *joint pdf*) is defined *f _{X,Y}*(

*x,y*), where.

*f*(_{X,Y}*x,y*) =*P*({*s*∈*S*\vert*X*(*s*) = [math]\displaystyle{ x }[/math],*Y*(*s*) =*y*}\})*f*(_{X,Y}*x,y*) =*P*(*X*=*x*,*Y*=*y*)

(b) Suppose that [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are two continuous random variables defined over the sample space *S*. The joint pdf of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], *f _{X,Y}*(

*x,y*), is the surface having the property that for any region [math]\displaystyle{ R }[/math] in the

*xy*-plane,

*P*((*X,Y*)∈*R*) =*P*({*s*∈*S*\vert (*X*(*s*). “Y*(*s*))∈*R*})**P*((*X,Y*)∈*R*) = Integral_{R}Integral [math]\displaystyle{ f }[/math]_{X,Y}(*x,y*)*dx dy*.

**Definition 3.3.2.**Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be two random variables defined on the same sample space *S*. The *joint cumulative distribution function* (or *joint cdf*) *of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math]* is defined [math]\displaystyle{ F_{X,Y}(x,y) }[/math], where

- [math]\displaystyle{ F_{X,Y}(x,y) }[/math] =
*P*({*s*∈*S*}*X*(*s*) <= [math]\displaystyle{ x }[/math] and*Y*(*s*) <=*y*}) - [math]\displaystyle{ F_{X,Y}(x,y) }[/math] =
*P*(*X*≤*x*,*Y*≤*y*).

}},

Author | volume | Date Value | title | type | journal | titleUrl | doi | note | year | |
---|---|---|---|---|---|---|---|---|---|---|

1986 AnIntroToMathStats | Richard J. Larsen Morris L. Marx | An Introduction to Mathematical Statistics and Its Applications, 2nd edition | http://books.google.com/books?id=AdinQgAACAAJ | 1986 |