Download III. Probability Theory

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ECONOMICS 4630/5630: REVIEW FOR MIDTERM I (Covers Topics 1-4)
General:
The test will consist of two sorts of problems. First will be a series of
statements to which you will be asked to respond, deciding whether each
statement is true, false, or uncertain. You must explain your reasoning in
order to get any credit. The second section will involve problems much like
those on your homework sets (see also problems in the text). You should be
able to not only solve problems, but demonstrate an understanding of what
you are doing and why. I will provide you with a formula sheet (see
attached), so there is no need to memorize formulae. Bring a calculator that
is not programmable; make sure its batteries are fresh. NOTE: cell phones
cannot be turned on for any reason during the test.
I.
Introduction
A.
Probability and Statistics
1.
What is probability and what is statistics?
2.
What is probability and statistics good for?
B.
Types of Data
C.
Types of Variables
D.
Levels of Measurement
E.
Terminology
II.
Descriptive Statistics
A.
Location or central tendency: Using Graphs
B.
Location or Central Tendency: Numerical Methods
1.
Mode
2.
Median and other percentiles
3.
Mean
C.
The Spread of a Distribution
1.
range
2.
interquartile range (IQR)
3.
variance and standard deviation (population)
4.
variance and standard deviation (sample)
III.
Probability Theory
A.
What is Probability in General?
B.
Probabilities of More Complex Events
1.
Probability Trees
2.
Outcome Sets
C.
Combinations of Events
1.
Union
2.
Intersection
3.
Complements
D.
Conditional Probability
E.
Independence
F.
Joint Distributions
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IV.
Discrete Probability Distributions
A.
Discrete probability distributions in general
B.
The uniform distribution
C.
The binomial (or Bernoulli)
D.
The hypergeometric distribution
E.
The Poisson distribution
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FORMULA SHEET
To find percentiles: grouped data
q th percentile  L 
where: L =
q=
q n  CF (i)
f
the lower limit of the class containing the percentile of interest
the percentile of interest, stated in decimal terms (e.g. 75th percentile would
be 0.75)
n=
total number of frequencies
f=
frequency in the class containing the percentile of interest
CF = cumulative number of frequencies in the classes preceding the
class containing the percentile of interest
i=
class interval
To find percentiles (raw data)
Position of qth percentile = (n  1)
Q
100
, where Q is the percentile of interest stated in
percent terms (i.e. 75th percentile would be 75)
2k rule
When grouping data, choose the smallest number, k, such that 2k > n, where n is the
sample size.
Rule for Determining Class Interval
i
H L
, where I is the class interval, H and L are the largest and smallest observations,
k
and k is the number of classes
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Complements
If A is the complement of A, then P A 1  PA 
Interquartile Range
IQR = Q3 - Q1, where Q1 is the 25th percentile and Q3 is the 75th percentile
Linear Combinations of Random Variables
If Y  a  bX , then  Y  a  b X and  Y  b 
X
Special rule of multiplication
If A, B, C, … , Z are events, assuming that each outcome is independent of every
other (that is, the occurrence of one outcome has no effect on the probability of
the occurrence of any other outcome), then P(A B  C …  Z) =
P(A)*P(B)*P(C)*…*P(Z).
Unions and Intersections
P( X  Y )  P( X )  P(Y )  P( X  Y )
This is the “general rule of addition”
If X and Y are mutually exclusive, then
P( X  Y )  P( X )  P(Y )
This is the “special rule of addition”
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Conditional Probability
P( X Y ) 
P( x y ) 
P( X  Y )
, or using probability distribution notation
P(Y )
P ( x, y )
P( y )
These are the “general rule of multiplication”
Independence
Using set notation, two events, X and Y, are independent if
P( X  Y )  P( X ) P(Y ) or if
P( X Y )  P( X )
Using probability distribution notation, X and Y are independent if
P(x,y) = P(x)P(y) for all x,y or if
P( x y)  P( x) for all x, y
Mean, Variance, and Standard Deviation (population formulae)
   xP( x)
 2   ( x   ) 2 P( x) or
 2   x 2 P( x)   2
  2
Mean, Variance, and Standard Deviation (sample formulae for raw data)
X 
1
 Xi
n
S2 
1
X i  X 2

n 1
S  S2
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Mean, Variance, and Standard Deviation (sample formulae for grouped data)
X
1 J
 X j f j , where j = 1, 2, …, J is the class number, Xj is the midpoint of
n j 1
class j, and fj is the frequency in class j
S2 


1 J
2
where j = 1, 2, …, J is the class number, Xj is the
 fj Xj X
n  1 j 1
midpoint of class j, and fj is the frequency in class j
S  S2
Uniform Probability Distribution
P x  
1
,
b  a 1
where a and b are the minimum and maximum values,
respectively.
Mean and variance of uniform

ab
2
2 
b  a b  a  2
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Binomial Probability Distribution
 n
P( x)    x (1   ) n  x
 x
Where:  = probability of success
n = # of trials
X = # of successes in n trials
 n
n!
n(n  1)( n  2)...(1)
  

 x  x!(n  x)! [ x( x  1)( x  2)...(1)][( n  x)( n  x  1)( n  x  2)...(1)
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Mean and variance of binomial
 X  n
 2 X  n (1   )
Hypergeometric Probability Distribution
S!
( N  S )!
X !( S  X )! (n  X )![( N  S )  (n  X )]!
P( X ) 
N!
n!( N  n)!
where
S = number of successes in population
n = sample size (# of trials)
N = population size
N-S = # of failures in the population
X = number of successes in the sample
Mean and variance of hypergeometric
x 
 x2 
nS
N
n  S  N  S 
N2

N  n 
N 1
Poisson Probability Distribution
P( X ) 
 x e 
x!
,
where e = 2.7183
X = # of successes
 = average (mean) number of successes
Mean and variance of Poisson
x = 
2x = 
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Binomial Coefficients
 n
n!
  
 x  x!(n  x)!
x
0
1
2
3
4
5
6
7
8
9
10
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
1
1
7
21
35
35
21
7
1
1
8
28
56
70
56
28
8
1
1
9
36
84
126
126
84
36
9
1
1
10
45
120
210
252
210
120
45
10
1
1
11
55
165
330
462
462
330
165
55
11
1
12
66
220
495
792
924
792
495
220
66
1
13
78
286
715
1,287
1,716
1,716
1,287
715
286
1
14
91
364
1,001
2,002
3,003
3,432
3,003
2,002
1,001
1
15
105
455
1,365
3,003
5,005
6,435
6,435
5,005
3,003
1
16
120
560
1,820
4,368
8,008
11,440
12,870
11,440
8,008
1
17
136
680
2,380
6,188
12,376
19,448
24,310
24,310
19,448
1
18
153
816
3,060
8,568
18,564
31,824
43,758
48,620
43,758
1
19
171
969
3,876
11,628
27,132
50,388
75,582
92,378
92,378
1
20
190
1,140
4,845
15,504
38,760
77,520
125,970
167,960
184,756
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Note: 0! = 1