重學Statistics, Cha5 Discrete Probabiliry Distributions

这一章学什么？

5.1 Random Variable

A random variable is a numerical description of the outcome of an experiment.

Discrete Random Variable: A random variable that may assume either a finite number of values or an infinite sequence of values such as 0, 1, 2, . . . is referred to as a discrete random variable. 例如：抛一个硬币可能出现的结果

Continuous Random Variable: A random variable that may assume any numerical value in an interval or collection of intervals is called a continuous random variable. 例如客户到店的时间。

5.2 Discrete Probability Distribution

一个汽车店的销售

Required Conditions for a Discrete Probability Function:  f(x) >= 0    ∑f(x) = 1

Discrete Uniform Probability Distribution:  f(x) = 1/n  where n= the number of values the random variable may have

5.3 Expected Value and Variance

Expected value of a discrete Random Varibale

E(x) = μ = ∑xf(x)

Variance of a Discrete Random Variable

Var(x) = σ²   =∑(x-μ)²f(x)

5.4 Binomial Probability Distribution

1. Properties of a Binomial experiment

1. The experiment consists of a sequence of nidenticaltrials.

2. Two outcomes are possible on each trial. We refer to one outcome as a success and the other outcome as a failure.

3. The probability of a success, denoted byp does not change from trial to trial. Consequently, the probability of a failure, denoted by 1-p, does not change from trial to trial.

4. The trials areindependent.

The probability distribution associated with this random variable is called the binomial probability distribution. 

2 Martin Clothing Store Problem

Let us consider the purchase decisions of the next three customers who enter the Martin Clothing Store. On the basis of past experience, the store manager estimates the probability that any one customer will make a purchase is .30. What is the probability that two of the next three customers will make a purchase? 

可能出现的结果：C3 2 =3

所以实验结果是两次成功，一次失败的概率是 pp(1-p) = 0.063

那么如果再算上三次实验的话，应该是 3* 0.063 = 0.189

Expect Value for Binomial Distribution: E(x) = μ = np

Variance for the Binomial Distribution: Var(x) =σ² = np(1-p)

5.5 Poisson Probability Distribution

1. PROPERTIES OF A POISSON EXPERIMENT

1. The probability of an occurrence is the same for any two intervals of equal length.

2. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.

2. An Example of Time Intervals

Suppose that we are interested in the number of arrivals at the drive-up teller window of a bank during a 15-minute period on weekday mornings. If we can assume that the probability of a car arriving is the same for any two time periods of equal length and that the arrival or nonarrival of a car in any time period is independent of the arrival or nonarrival in any other time period, the Poisson probability function is applicable. Suppose these assumptions are satisfied and an analysis of historical data shows that the average number of cars arriving in a 15-minute period of time is 10. What is f(5)?

满足这两个条件，第一在两段相等的时间，来的车辆数是相等的，第二，车来和不来的可能性是独立的，15分钟来5辆车的可能性是多少？

直接套用 poisson probability function的公式来做：

3. Hyoereometric Probability Distribution

The hypergeometric probability distribution is closely related to the binomial distribution. The two probability distributions differ in two key ways. With the hypergeometric distribution, the trials are not independent; and the probability of success changes from trial to trial.

The hypergeometric probability function is used to compute the probability that in a random selection of n elements, selected without replacement, we obtain x elements labeled success and n- x elements labeled failure. For this outcome to occur, we must obtain x successes from the r successes in the population and n - x failures from the N - r failures.

与 binomial distribution不同的是，这个的概率会改变。用来计算任意选择的可能性，在n中选择了不放回去，x算成功，n-x失败

Example：

Electric fuses produced by Ontario Electric are packaged in boxes of 12 units each. Suppose an inspector randomly selects three of the 12 fuses in a box for testing. If the box contains exactly five defective fuses, what is the probability that the inspector will find exactly one of the three fuses defective? In this application, n = 3 and N = 12. With r = 5 defective fuses in the box the probability of finding x = 1 defective fuse is

一个盒子有12个产品，5个坏的，从中任意抽三个，正好有一个是坏的的可能性？