Презентация Discrete Probability Distributions: Binomial and Poisson Distribution. Week 7 (2) онлайн
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Презентации » Математика » Discrete Probability Distributions: Binomial and Poisson Distribution. Week 7 (2)
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- Тип файла:ppt / pptx (powerpoint)
- Всего слайдов:62 слайда
- Для класса:1,2,3,4,5,6,7,8,9,10,11
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Слайды и текст к этой презентации:
№3 слайд
![Probability and cumulative](/documents_6/1ade573e87197b642fa179b29979bbf0/img2.jpg)
Содержание слайда: Probability and cumulative probability
distribution of a discrete random variable
In the last class we saw how to calculate the probability
of a specific discrete random variable, such as P( x = 3)
and the cumulative probability such as P(x 3) in n trials
with the following formula:
We need to know n, x and P (probability of success)
№5 слайд
![Probability and cumulative](/documents_6/1ade573e87197b642fa179b29979bbf0/img4.jpg)
Содержание слайда: Probability and cumulative probability
distribution of a discrete random variable
Ali, real estate agent has 5 potential customers to buy a house or apartment with a probability of 0.40. The probability and cumulative probability table of the sale of houses here below. We are able to answer what is P( x=3), P(x 4), P(x >3)etc.
№7 слайд
![Binomial Distribution shapes](/documents_6/1ade573e87197b642fa179b29979bbf0/img6.jpg)
Содержание слайда: Binomial Distribution shapes
When P = .5 the shape of the distribution is perfectly symmetrical and resembles a bell-shaped (normal distribution)
When P = .2 the distribution is skewed right. This skewness increases as P becomes smaller.
When P = .8, the distribution is skewed left. As P comes closer to 1, the amount of skewness increases.
№9 слайд
![Solving Problems with](/documents_6/1ade573e87197b642fa179b29979bbf0/img8.jpg)
Содержание слайда: Solving Problems with Binomial Tables
MSA Electronics is experimenting with the manufacture of a new USB-stick and is looking into the number of defective USB-sticks
Every hour a random sample of 5 USB-sticks is taken
The probability of one USB-stick being defective is 0.15
What is the probability of finding 3, 4, or 5 defective USB-sticks ?
P( x = 3), P(x = 4 ), P(x= 5)
№11 слайд
![Solving Problems with](/documents_6/1ade573e87197b642fa179b29979bbf0/img10.jpg)
Содержание слайда: Solving Problems with Binomial Tables
MSA Electronics is experimenting with the manufacture of a new USB-stick
Every hour a random sample of 5 USB-sticks is taken
The probability of one USB-stick being defective is 0.15
What is the probability of finding more than 3 (3 inclusive) defective USB-sticks? P( x 3)
№20 слайд
![Poisson Random Variable -](/documents_6/1ade573e87197b642fa179b29979bbf0/img19.jpg)
Содержание слайда: Poisson Random Variable - three requirements
1. The number of expected outcomes in one interval of time or unit space is unaffected (independent) by the number of expected outcomes in any other non-overlapping time interval.
Example: What took place between 3:00 and 3:20 p.m. is not affected by what took place between 9:00 and 9:20 a.m.
№21 слайд
![Poisson Random Variable -](/documents_6/1ade573e87197b642fa179b29979bbf0/img20.jpg)
Содержание слайда: Poisson Random Variable - three requirements (continued)
2.The expected (or mean) number of outcomes over any time period or unit space is proportional to the size of this time interval.
Example:
We expect half as many outcomes between 3:00 and 3:30 P.M. as between 3:00 and 4:00 P.M.
3.This requirement also implies that the probability of an occurrence must be constant over any intervals of the same length.
Example:
The expected outcome between 3:00 and 3:30P.M. is equal to the expected occurrence between 4:00 and 4:30 P.M..
№22 слайд
![Examples of a Poisson Random](/documents_6/1ade573e87197b642fa179b29979bbf0/img21.jpg)
Содержание слайда: Examples of a Poisson Random variable
The number of cars arriving at a toll booth in 1 hour (the time interval is 1 hour)
The number of failures in a large computer system during a given day (the given day is the interval)
The number of delivery trucks to arrive at a central warehouse in an hour.
The number of customers to arrive for flights at an airport during each 10-minute time interval from 3:00 p.m. to 6:00 p.m. on weekdays
№23 слайд
![Situations where the Poisson](/documents_6/1ade573e87197b642fa179b29979bbf0/img22.jpg)
Содержание слайда: Situations where the Poisson distribution
is widely used: Capacity planning – time interval
Areas of capacity planning observed in a sample:
A bank wants to know how many customers arrive at the bank in a given time period during the day, so that they can anticipate the waiting lines and plan for the number of employees to hire.
At peak hours they might want to open more guichets (employ more personnel) to reduce waiting lines and during slower hours, have a few guichets open (need for less personnel).
№25 слайд
![Example Drive-up ATM Window](/documents_6/1ade573e87197b642fa179b29979bbf0/img24.jpg)
Содержание слайда: Example: Drive-up ATM Window
Poisson Probability Function: Time Interval
Suppose that we are interested in the number of arrivals at the drive-up ATM window of a bank during a 15-minute period on weekday mornings.
If we assume that the probability of a car arriving is the same for any two time periods of equal length and that the arrival or non-arrival of a car in any time period is independent of the arrival or non-arrival in any other time period, the Poisson probability function is applicable.
Then if we assume that an analysis of historical data shows that the average number of cars arriving during a 15-minute interval of time is 10, the Poisson probability function with = 10 applies.
№33 слайд
![Continuous random variable A](/documents_6/1ade573e87197b642fa179b29979bbf0/img32.jpg)
Содержание слайда: Continuous random variable
A continuous random variable can assume any value in an interval on the real line or in a collection of intervals.
It is not possible to talk about the probability of the random variable assuming a particular value, because the probability will be close to 0.
Instead, we talk about the probability of the random variable assuming a value within a given interval.
№35 слайд
![Probability Density Function](/documents_6/1ade573e87197b642fa179b29979bbf0/img34.jpg)
Содержание слайда: Probability Density Function
The probability density function, f(x), of a continuous random variable X has the following properties:
f(x) > 0 for all values of x
The area under the probability density function f(x) over all values of the random variable X within its range, is equal to 1.0
The probability that X lies between two values is the area under the density function graph between the two values
№36 слайд
![Probability Density Function](/documents_6/1ade573e87197b642fa179b29979bbf0/img35.jpg)
Содержание слайда: Probability Density Function
The probability density function, f(x), of random variable X
has the following properties:
The cumulative density function F(x0) is the area under the
probability density function f(x) from the minimum x value
up to x0
where xm is the minimum value of the random variable x
№46 слайд
![The Normal Distribution](/documents_6/1ade573e87197b642fa179b29979bbf0/img45.jpg)
Содержание слайда: The Normal Distribution
Symmetrical with the midpoint representing the mean
Shifting the mean does not change the shape
Values on the X axis are measured in the number of standard
deviations away from the mean 1 2 3
As standard deviation becomes larger, curve flattens
As standard deviation becomes smaller, curve becomes steeper
№50 слайд
![The Standard Normal](/documents_6/1ade573e87197b642fa179b29979bbf0/img49.jpg)
Содержание слайда: The Standard Normal Distribution – z-values
Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z), with mean 0 and standard deviation 1
We say that Z follows the standard normal distribution.
№55 слайд
![Haynes Construction Company](/documents_6/1ade573e87197b642fa179b29979bbf0/img54.jpg)
Содержание слайда: Haynes Construction Company
Builds three- and four-unit apartment buildings:
Total construction time follows a normal distribution
For triplexes, m = 100 days and = 20 days
Contract calls for completion in 125 days
Late completion will incur a severe penalty fee
Probability of completing in 125 days? P(x <125)
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