概率论与数理统计英文版第三章 - 图文

Chapter 3. Random Variables and Probability Distribution

1. Concept of a Random Variable

Example: three electronic components are tested sample space (N: non defective, D: defective)

S =｛NNN, NND, NDN, DNN, NDD, DND, DDN, DDD｝ allocate a numerical description of each outcome concerned with the number of defectives

each point in the sample space will be assigned a numerical value of 0, 1, 2, or 3. random variable X: the number of defective items, a random quantity

random variable Definition 3.1

A random variable is a function that associates a real number with each element in the sample space. X: a random variable x : one of its values

Each possible value of X represents an event that is a subset of the sample space electronic component test:

E =｛DDN, DND, NDD｝ =｛X = 2｝.

Example 3.1 Two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. Y is the number of red balls. The possible outcomes and the values y of the random variable Y ?

Example 3.2 A stockroom clerk returns three safety helmets at random to three steel mill employees who had previously checked them. If Smith, Jones, and Brown, in that order, receive one of the three hats, list the sample points for the possible orders of returning the helmets,and find the value m of the random variable M that represents the number of correct matches.

The sample space contains a finite number of elements in Example 3.1 and 3.2. another example: a die is thrown until a 5 occurs, F: the occurrence of a 5 N: the nonoccurrence of a 5

obtain a sample space with an unending sequence of elements S =｛F, NF, NNF, NNNF, . . .｝

the number of elements can be equated to the number of whole numbers; can be counted

Definition 3.2 If a sample space contains a finite number of possibilities or an unending sequence with as many elements as there are whole numbers, it is called a discrete sample space.

The outcomes of some statistical experiments may be neither finite nor countable.

example: measure the distances that a certain make of automobile will travel over a prescribed test course on 5 liters of gasoline

distance: a variable measured to any degree of accuracy

we have infinite number of possible distances in the sample space, cannot be equated to the number of whole numbers.

Definition 3.3

If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space

A random variable is called a discrete random variable if its set of possible outcomes is countable. Y in Example 3.1 and M in Example 3.2 are discrete random variables.

When a random variable can take on values on a continuous scale, it is called a continuous random variable.

The measured distance that a certain make of automobile will travel over a test course on 5 liters of gasoline is a continuous random variable.

continuous random variables represent measured data:

all possible heights, weights, temperatures, distance, or life periods.

discrete random variables represent count data: the number of defectives in a sample of k items, or the number of highway fatalities per year in a given state.

2. Discrete Probability Distribution

A discrete random variable assumes each of its values with a certain probability

assume equal weights for the elements in Example 3.2, what's the probability that no employee gets back his right helmet. The probability that M assumed the value zero is 1/3. The possible values m of M and their probabilities are 0 1 3 1/3 1/2 1/6

Probability Mass Function

It is convenient to represent all the probabilities of a random variable X by a formula. write p(x) = P (X = x)

The set of ordered pairs (x, p(x)) is called the probability function or probability distribution of the discrete random variable X.

Definition 3.4

The set of ordered pairs (x, p(x)) is a probability function, probability mass function, or probability distribution of the discrete random variable X if, for each possible outcome x

Example 3.3 A shipment of 8 similar microcomputers to a retail outlet contains 3 that are defective. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives. Solution

X: the possible numbers of defective computers x can be any of the numbers 0, 1, and 2.

Cumulative Function

There are many problem where we may wish to compute the probability that the observed value of a random variable X will be less than or equal to some real number x. Writing F (x) = P (X≤x) for every real number x.

Definition 3.5

The cumulative distribution F (x) of a discrete random variable X with probability distribution p(x) is

For the random variable M, the number of correct matches in Example 3.2, we have

The cumulative distribution of M is

Remark. the cumulative distribution is defined not only for the values assumed by given random variable but for all real numbers.

Example 3.5 The probability distribution of X is

Find the cumulative distribution of the random variable X.

Certain probability distribution are applicable to more than one physical situation. The probability distribution of Example 3.5 can apply to different experimental situations. Example 1: the distribution of Y , the number of heads when a coin is tossed 4 times

Example 2: the distribution of W , the number of read cards that occur when 4 cards are drawn at random from a deck in succession with each card replaced and the deck shuffled before the next drawing.

graphs

It is helpful to look at a probability distribution in graphic form. bar chart; histogram;

cumulative distribution.

3. Continuous Probability Distribution

Continuous Probability distribution

A continuous random variable has a probability of zero of assuming exactly any of its values. Consequently, its probability distribution cannot be given in tabular form.

Example: the heights of all people over 21 years of age (random variable)

Between 163.5 and 164.5 centimeters, or even 163.99 and 164.01 centimeters, there are an infinite number of heights, one of which is 164 centimeters.

The probability of selecting a person at random who is exactly 164 centimeters tall and not one of the infinitely large set of heights so close to 164 centimeters is remote.

We assign a probability of zero to a point, but this is not the case for an interval. We will deal with an interval rather than a point value, such as P (a < X < b), P (W≥ c).

P (a≤X≤b) = P (a < X≤ b) = P (a≤X < b) = P (a < X < b)

where X is continuous. It does not matter whether we include an endpoint of the interval or not. This is not true when X is discrete.

Although the probability distribution of a continuous random variable cannot be presented in tabular form, it can be stated as a formula.

refer to histogram

Definition 3.6 The function f(x) is a probability density function for the continuous random variable X, defined over the set of real numbers R, if

Example 3.6 Suppose that the error in the reaction temperature, in oC, for a controlled laboratory experiment is a continuous random variable X having the probability density function

(a) Verify condition 2 of Definition 3.6. (b) Find P (0 < X≤ 1).

Solution: . . . . . . P (0 < X≤1) = 1/9.

Definition 3.7 The cumulative distribution F (x) of a continuous random variable X with density function f(x) is

immediate consequence:

Example 3.7 For the density function of Example 3.6 find F (x), and use it to evaluate P (0 < x≤1).

4. Joint Probability Distributions

the preceding sections: one-dimensional sample spaces and a single random variable situations: desirable to record the simultaneous outcomes of several random variables.

Joint Probability Distribution

Examples

1. we might measure the amount of precipitate P and volume V of gas released from a controlled chemical experiment; we get a two-dimensional sample space consisting of the outcomes (p, v).

2. In a study to determine the likelihood of success in college, based on high school data, one might use a three-dimensional sample space and record for each individual his or her aptitude test score, high school rank in class, and grade-point average at the end of the freshman year in college.

X and Y are two discrete random variables, the joint probability distribution of X and Y is p (x, y) = P (X = x, Y = y)

the values p(x, y) give the probability that outcomes x and y occur at the same time.

Definition 3.8 The function p(x, y) is a joint probability distribution or probability mass function of the discrete random variables X and Y if

Example 3.8

Two refills for a ballpoint pen are selected at random from a box that contains 3 blue refills,2 red refills, and 3 green refills. If X is the number of blue refills and Y is the number of red refills selected, find (a) the joint probability function p(x, y)

(b) P [(X, Y )∈A] where A is the region｛(x, y)|x + y≤1｝ Solution

the possible pairs of values (x, y) are (0, 0), (0, 1), (1, 0), (1, 1), (0, 2), and (2, 0).

p (x, y) represents the probability that x blue and y red refills are selected.

(b) P [(X, Y )∈A] = 9/14

present the results in Table 3.1

Definition 3.9 The function f(x, y) is a joint density function of the continuous random variables X and Y if

When X and Y are continuous random variables, the joint density function f(x, y) is a surface lying above the xy plane.

P [(X, Y )∈ A], where A is any region in the xy plane, is equal to the volume of the right cylinder bounded by the base A and the surface.

Example 3.9 Suppose that the joint density function is

(b) P [(X, Y )∈A]= 13/160

marginal distribution

p (x, y): the joint probability distribution of the discrete random variables X and Y

the probability distribution p X(x) of X alone is obtained by summing p(x, y) over the values of Y .

Similarly, the probability distribution p Y (y) of Y alone is obtained by summing p(x, y) over the values of X. pX (x) and p Y (y): marginal distributions of X and Y

When X and Y are continuous random variables, summations are replaced by integrals.

Definition 3.10 The marginal distribution of X alone and of Y alone are

Example 3.10 Show that the column and row totals of Table 3.1 give the marginal distribution of X alone and of Y alone.

Example 3.11 Find marginal probability density functions fX(x) and fy(y)for the joint density function of Example 3.9.

The marginal distribution pX(x) [or fX(x)] and px(y) [or fy(y)] are indeed the probability distribution of the individual variable X and Y , respectively. How to verify?

The conditions of Definition 3.4 [or Definition 3.6] are satisfied.

Conditional distribution

recall the definition of conditional probability:

X and Y are discrete random variables, we have

The value x of the random variable represent an event that is a subset of the sample space.

Definition 3.11

Let X and Y be two discrete random variables. The conditional probability mass function of the random variable Y , given that X = x, is

Similarly, the conditional probability mass function of the random variable X, given that Y = y, is

Definition 3.11'

Let X and Y be two continuous random variables. The conditional probability density function of the random variable Y , given that X = x, is

Similarly, the conditional probability density function of the random variable X, given that Y = y, is

Remark:

The function f(x, y)/fX(x) is strictly a function of y with x fixed, the function f(x, y)/fy(y) is strictly a function of x with y fixed, both satisfy all the conditions of a probability distribution.

How to find the probability that the random variable X falls between a and b when it is known that Y = y

Example 3.12 Referring to Example 3.8, find the conditional distribution of X, given that Y = 1, and use it to determine P (X = 0|Y = 1).

Example 3.13 The joint density for the random variables (X, Y ) where X is the unit temperature change and Y is the proportion of spectrum shift that a certain atomic particle produces is

(a)Find the marginal densities fX(x), fy(y), and the conditional density fY??X (y?x)

(b)Find the probability that the spectrum shifts more than half of the total observations, given the temperature is increased to 0 .25 unit. (a)

(b)

Example 3.14 Given the joint density function

(a) (b)

statistical independence

events A and B are independent, if

P (B∩A) = P (A)P (B).

discrete random variables X and Y are independent, if P (X = x, Y = y) = P (X = x)P (Y = y) for all (x, y) within their range.

The value x of the random variable represent an event that is a subset of the sample space.

Definition 3.12 Let X and Y be two discrete random variables, with joint probability distribution p(x, y) and marginal distributions pX(x)and pY (y), respectively. The random variables X and Y are said to be statistically independent if and only if

p (x,y) = pX(x)pY (y) for all (x, y) within their range.

Definition 3.12' Let X and Y be two continuous random variables, with joint probability distribution f(x, y) and marginal distributions fX(x) and fY (y), respectively. The random variables X and Y are said to be statistically independent if and only if

f (x, y) =fX(x)fY (y) for all (x, y) within their range.

The continuous random variables of Example 3.14 are statistically independent. However, that is not the case for the Example 3.13.

For discrete variables, requires more thorough investigation. If you find any point (x, y) for which p(x, y) is defined such that p(x, y)

≠pX(x)pY (y), the discrete variables X and Y are not statistically independent. Example 3.15 Show that the random variables of Example 3.8 are not statistically independent.

the case of n random variables

joint marginal distributions of two r.v. X1 and X2

Definition 3.13 Let x1, x2,… , xn be n discrete random variables, with joint probability distribution p(x1, x2,… , xn)

and marginal distributions pX1 (x1), pX2 (x2), …, pXn (xn),respectively. The random variables x1, x2,… , xn are mutually statistically independent,then

for all (x1, x2,… , xn) within their range.

Definition 3.13' Let x1, x2,… , xn be n continuous random variables, with joint probability distribution f(x1, x2,… , xn)

and marginal distributions fX1 (x1), fX2 (x2), …, fXn (xn)respectively. The random variables x1, x2,… , xn are mutually statistically independent, then

for all(x1, x2,… , xn)within their range.

Example 3.16 Suppose that the shelf life , in years, of a certain perishable food product packaged in cardboard containers is a random variable whose probability density function is given by

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