Chapter 3. Random Variables and Probability Distribution
笔记本维修学校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
11月英文缩写Each possible value of X reprents an event that is a subt 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 reprents 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 quence of elements
S =｛F, NF, NNF, NNNF, . . .｝
the number of elements can be equated to the number of whole numbers; can be counted
kubo人的肢体语言Definition 3.2 If a sample space contains a finite number of possibilities or an unending quence 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 cour on 5 liters of gasoline
riskydistance: a variable measured to any degree of accuracy 软件开发培训班
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 gment, it is called a continuous sample space
A random variable is called a discrete random variable if its t 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 cour on 5 liters of gasoline is a continuous random variable.
棺椁怎么读continuous random variables reprent measured data:
all possible heights, weights, temperatures, distance, or life periods.
discrete random variables reprent 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 reprent all the probabilities of a random variable X by a formula.
write p(x) = P (X = x)
The t of ordered pairs (x, p(x)) is called the probability function or probability distribution of the discrete random variable X.
Definition 3.4
The t 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 purcha of 2 of the computers, find the probability distribution for the number of defectives. 壁毯
Solution
X: the possible numbers of defective computers
x can be any of the numbers 0, 1, and 2.
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