Download probability distribution

Principles of Statistics and Economics Statistics
2014 Dongdong Ge
Principles of Statistics and Economics Statistics
SHUFE
Instructor: Dongdong Ge
Case in Practice
•
CitiBank in Long Island City
x
0
probability
.1353
1
.2707
2
.2707
3
.1804
4
.0902
>=5
.0527
Learning Objectives
• Discrete Probability distribution
• Expected Value (Mean) and Variance
• Binomial Probability Distribution
• Poisson Probability Distribution
• Hypergeometric Probability Distribution
Four steps in deciding the
probability for an event
• Identify experimental outcomes
• Assigning probabilities
• Define the event of interest
• Calculating the probability for the event
Example
• An automobile dealer in New York
• Over the past 300 days of operation:
–
–
–
–
–
–
54 days with no automobiles sold;
117 days with one automobile sold;
72 days with two automobiles sold;
42 days with three automobiles sold;
12 days with four automobiles sold;
3 days with five automobiles sold;
• Question: What is the probability that at least
one automobile is sold in a day?
Random variable (随机变量)
• Random variable: a numerical description of the
outcome of an experiment.
• It must assume numerical values.
• Discrete random variables (离散型随机变量):
assume either a finite number of values or an
infinite sequence of values such as 0, 1, 2, ……
• Continuous random variables (连续型随机变量):
assume any numerical value in an interval or
collection of intervals
Examples of random variables
Experiment Random variable
Possible values
for the random
variable
Operate a
restaurant
for a day
Number of
customers
0, 1, 2, 3, ……
Time between
customer arrivals
in minutes
X>=0
Probability distribution
• The probability distribution (概率分布) of a
random variable describes how probabilities are
distributed over the values of the random
variables.
• The function that defines the probability
distribution is called probability function (概率
方程).
• Advantage: once a distribution is known, it is
relatively easy to determine the probability of a
variety of events.
An example
1
2
3
4
1/10
2/10
3/30
4/10
What’s the probability function of it?
Expected Value: Mean (数学期望)
• A measure of the central location for a random
variable.
– Expected value or the mean of the random variable X
is defined as;
E( x)     xf ( x)
Variance (方差)
• Spread of the data from the center (mean)
• The average of the squared deviations from
the mean
Var ( x)   ( x   ) f ( x)
2
2
Var ( x)   E ( x )  E ( x)
2
2
2
Binomial probability distribution (二
项概率分布)
• Properties of a Binomial Experiment
– A sequence of n identical trials;
– Two outcomes are possible for each trial:
success and failure;
– The probability of success, denoted by p,
does not change from trial to trial;
– The trials are independent.
Example
• Consider a clothing store, the store
manager wants to know the probability
that two of the next three customers will
make a purchase.
• On the basis of past experience, the store
manager estimates the probability that any
one customer will make a purchase is p.
Binomial probability distribution
• Probability function
• Expected value
• Variance
n x
f ( x)    p (1  p)n x
 x
E ( x)    np
Var ( x)   2  np(1  p)
Poisson probability distribution (泊
松概率分布)
• Useful in estimating the number of
occurrences over a specified internal of
time or space.
• Two properties:
– The probability of an occurrence is the same
for any two intervals of equal lengths;
– The occurrence / nonoccurence in any interval
is independent of the occurrence /
nonoccurence in any other interval.
Poisson probability distribution
• Distribution function
f ( x) 
• Expected value
 xe 
x!
E ( x)  
• Variance
Var ( x)  
Example
• # of arrivals at the drive-up teller window of a bank
during a 15-minute period.
• The average # of arrivals=10.
10 x e 10
f ( x) 
x!
• What’s the probability of exact 5 arrivals in 15 mins?
• What’s the probability of exact 1 arrival in a 3minute period?
2 x e 2
f ( x) 
x!
Back to CityBand Case
• What’s expectation?
• What’s probability
function?
2
x
0
1
2
3
4
>=5
probability
.1353
.2707
.2707
.1804
.0902
.0527
Hypergeometric Probability
Distribution (超几何概率分布)
• N the size of population, of which r labeled
success
• n is selected, of which x labeled success.
• Probability function: probability of x successes in
n trials.
 r  N  r 
 

x
n

x

f ( x)    
N
 
n
Hypergeometric Probability
Distribution
• Example:
– An inspector randomly selects 3 of the 12
products for testing. If the box contains
exactly 5 defective products, what is the
probability that the inspector will find exactly 1
of the 3 products defective.
Hypergeometric Probability
Distribution
r
E ( x )    n( )
N
 r  r  N  n 
Var ( x)    n  1  

 N  N  N  1 
2
Any other discrete distribution?
• Uniform
• Geometric Distribution
• More…