Schaum

1

Special Probability Distributions

• Schaums Outline
• Probability and Statistics
• Chapter 4
• Presented by Carol Dahl

2
Chapter 4 Outline

• Binomial Distribution
• Normal Distribution
• Poisson Distribution
• Relations Between Distributions
• Binomial and Normal
• Binomial and Poisson
• Poisson and Normal
• Central Limit Theorem

3
Outline

• Multinomial Distribution
• Hypergeometric Distribution
• Uniform Distribution
• ?2 Distribution
• t-Distribution
• F-Distribution
• Cauchy
• Exponential
• Lognormal

4
Introduction

• Special Probability Distributions
• give probabilities for random variables
• discrete and continuous
• help us make inferences

5
Distributions Help Make Inferences

• Powerful tools - uncertainty
• prediction
• confidence intervals Q ßo ß1P
  ß2Y
• hypothesis tests

6
Binomial Distribution

• You own ten draglines for mining coal

7
Binomial (Bernoulli) Distribution

• Probability associated with no repairs
• P(no repairs) p
• P(repairs) (1-p) q
• If breakdown between machines independent
• Bernoulli Trial
• n trials 10
• x number with no repairs out of n draglines
• Binomial distribution
• P(Xx)?n? pxqn-xn!/(x!(n-x)!)pxqn-x
• ?x?

8
Binomial Example

• Dragline P(repairs) 0.2
• Binomial P(Xx) ( n ) px(1-p)n-x
• 
  ( x )
• P(X2) (10) 0.22 (1-0.2)10-2 10!
  0.220.88
• 2 2!(10-2)!
• 0.302
• Excel insert, function, statistical, binomdist
• binomdist(x,n,p,cumulative)
• (true or
  false)
• binomdist(2,10,0.2,false)

9
Probabilities of Dragline Repairs

• Number of Probability of
• repairs Repair

0 0.107
1 0.268
2 0.302
3 0.201
4 0.088
5 0.026
6 0.006
7 0.001
8 0.000
9 0.000
10 0.000
10
Probabilities of Dragline Repairs
11
Properties of Binomial Distribution

• Discrete
• Suppose n 10, P 0.2
• Mean ?np 100.2 8
• Variance ?2npq 100.20.8 1.6
• Standard deviation ? (1.6 )0.5 1.265
• Coefficient of skewness
• a3(q-p)/? (0.8 0.2)/ 1.265 0.474
• Coefficient of kurtosis
• a43(1-6pq)/npq 3 (1-60.80.2)/1.6 3.025

12
Functions Relating to Binomial

• Moment generating function M(t)(qpet)n
• M(t) E(etX) ?xetXP(X)
• E(X) ?xXP(X) M'(0)
• M'(t)n(qpet)n-1pet
• M'(0)n(qpe0)n-1pe0 n((1-p)p)n-1p np
• E(X2) ?xX2P(X) M''(0)
• E(X3) ?xX3P(X) M'''(0)
• Replacing t by i? with i imaginary number
• (-1)0.5 then we get another useful function
• Characteristic function f(?)(qpei?)n

13
Law of Large Numbers for Bernouilly Trials

• Estimate p by sampling p x/n
• By increasing number of trials
• can get as close as we want to true mean
• lim P (X/n pgte) 0
• n-gt?

14
Standard Normal (0,1)

• Most important continuous distribution (Gaussian)
• f(Z) 1 exp(-Z2/2) dZ
• (2?2)0.5

15
Standard Normal (0,1) Example

• Building a hydro Three Gorges 18,000 MW

16
Standard Normal (0,1)

• Want to know
• how much rainfall deviated from normal Z
• Z N(0,1) with Z measured in inches
• Five things you might want to know
• P(Z lt a) P(Z gt b)
• P(Z lt -c) P(Z gt -d)
• P(e lt Z lt f)

17
Standard Normal (0,1)

• f(Z) 1 exp(-Z2/2) dZ
• (2?2)0.5
• P(Z lt a) P(Z gt b)
• P(Z lt -c) P(Z gt -d)
• P(e lt Z lt f)

18
Standard Normal (0,1)

• P(Z lt a) 1 ? aexp(-Z2/2) dZ
• (2?)0.5 -?
• P(Z gt b) 1 ? ? exp(-Z2/2) dZ
• (2?)0.5 b
• P(Z lt -c) 1 ?-cexp(-Z2/2) dZ
• (2?)0.5 -?
• P(Z gt -d) 1 ? ? exp(-Z2/2) dZ
• (2?)0.5 -d
• P(e lt Z lt f) 1 ? f exp(-Z2/2) dZ
• (2?)0.5 e
• P(e lt Z lt f) P(Z ltf) P(Z lt e)

19
Standard Normal (0,1)

• But difficult to integrate
• use Tables, Excel, other computer packages
• Normal Tables Schaums GHJ
• P(0ltZltz) 1 ? z exp(-Z2/2) dZ
• (2?)0.5 0

20
Standard Normal (0,1) - Table

• Table P(0ltZltz)
• P(0ltZlt 2) 0.477
• P(Zlt2) 0.5 0.477

21
Standard Normal (0,1) - Table

• Table P(0ltZltz)
• (Zgt1.52)1P(Zlt 1.52)
• 1 (0.5 0.436) 0.564

22
Standard Normal (0,1) - Table

• Table P(0ltZltz)
• (Zgt-1.58)P(Zlt1.58)
• 0.5 0.443

23
Standard Normal (0,1) - Table

• Table P(0ltZltz)
• (Zlt-2.56)P(Zgt2.56)
• 1- P(Zlt2.56)
• (1(0.50.495)0.005

24
Standard Normal (0,1) - Table

• Table
• P(0ltZltz)gt
• (-0.54ltZlt2.02)
• P(Zlt2.02)-(Zlt-0.54)
• ?

25
Standard Normal (0,1) - Table

• Excel
• P(- ?ltZltz) normsdist(z,true)
• P(Zlt2.0) normsdist(2) 0.977

26
Standard Normal (0,1) - Table

• Excel
• P(- ?ltZltz) normsdist(z,true)
• P(Zgt2.0) 1- normsdist(2) 0.023
• P(Z gt -2) 1- normsdist(-2) 0.97

27
Inverse Normal

• Normal P(Zlt1) ??
• Deviation in rainfall lt 1 inch above normal
  what of the time?
• P(Zlt-1.5) ??
• Deviation in rainfall lt 1.5 inch below normal
• what of the time?
• Inverse Normal
• P(Zltz) 0.05
• Rain fall deviates lt what amount 5 of the time

28
Standard Normal (0,1) Inverse

• Table P(0ltZltz)gt
• P(Zlta) 0.698
• 0.5 0.198
• a 0.52

29
Standard Normal (0,1) - Inverse

• Table P(0ltZltz)gt
• P(Zgta) 0.476
• P(Zgta) 1 P(Zlta)
• P(Zlta) 0.524
• 0.5 0.024
• a 0.06

30
Standard Normal (0,1) - Inverse

• Table P(0ltZltz)gt
• P(Zlta) 0.288 P(Zgt-a)
• P(Zlt-a) 1 P(Zgt-a)
• 0.712 0.5 0.212
• -a 0.56 -gt a?

31
Standard Normal (0,1) - Inverse

• Table P(0ltZltz)gt
• P(Zgta) 0.846
• P(Zgta) P(Zlt-a)
• P(Zlt-a) 0.5 0.346
• - a 1.02
• a -1.02

32
Normal Example Inverse Excel

• P (Zlta) 0.698 normsinv(0.698) 0.519
• P (Zlta) 0.288 normsinv(0.288) -0.559
• P (Zgta) 0.846 -normsinv(0.846) -1.019
• P (Zgta) 0.210 normsinv(1-0.210) 0.806
• P(-altZlta) 0.5 normsinv(0.75) 0.67449

33
Properties of Normal Distribution

• Mean ?
• Variance ?2
• Standard deviation ?
• Coefficient of skewness ?30
• Coefficient of kurtosis ?43
• Moment generating function
  M(t)e? t(?2t2/2)
• Characteristic function ?(?)ei? ?-(? 2 ? 2/2)

34
Relation between Distributions

• Binomial gt Normal n gets large

35
Poisson Distribution

• Discrete infinite distribution with pdf
• f(x)P(Xx)?xe- ? /x! x0,1,2,3,...
• ? mean
• decay radioactive particles
• demands for services
• demands for repairs
•  

36
Poisson Distribution

•  
• Example X number of well workovers/month
• X poisson mean ? 5
• P(X 3) 53e-5 0.140
• 5!

37
Poisson Distribution
in Excel

• Excel
• poisson(x,?(mean),cumulative)
• true or
  false
• P(X 3)
• poisson(3,5,false) 0.14
• P(Xlt 3)
• poisson(3,5,true) 0.265
• p(X gt 8)
• 1 - poisson(7,5,true) 0.13

38
Properties of Poisson

• Mean ? ?
• Variance ?2 ?
• Standard deviation ? ?1/2
• Coefficient of skewness ? 3 ?-1/2
• Coefficient of kurtosis ? 4 3 1/?
• Moment generating function M(t)e? (et-1)
• Characteristic function ?(?)e? (e(i?)-1)

39
Relations between Distributions

• Poisson and Binomial close
• when n large p small
• Poisson and Normal are close when n gets large
• To standardize Poisson
• Z (X - ?)/ ?0.5

40
Central Limit Theorem
random variables X1, X2,   independent
identically distributed finite mean m and
variance s2. then ?X (X1 X2 Xn)/n goes
to a N(m, s2/n) as n -gt ?
41
Multinomial Distribution

• Example You work for a gas company
• likelihood a family will buy
• gas furnace is 1/2 (p1)
• electric furnace is 1/3 (p2)
• fuel oil furnace is 1/6 (p3)
• 10 furnaces (n) replaced in next heating season
  probability that
• 5 (x1) gas?
• 4 (x2) electric
• 1 (x3) fuel oil?

42
Multinomial Distribution
Generalization of binomial A1, A2, A3,Ak are
events occur with probabilities p1, p2,pk If
X1, X2, Xk are random variables number of
times that A1, A2,Ak occur n trials X1
X2Xk n then P(X1 n1, X2 n2,, nk)
n! p1n1 p2n2pknk
n1! n2!nk! Where n1 n2, nk n
43
Multinomial Distribution

• Work out probability for
• 5 (x1) gas?
• 4 (x2) electric
• 1 (x3) fuel oil?
• P(x15, x24, x31)

44

• Hypergeometric Distribution

Example Box of drilling bits contains 5 X
bit 4 bit 3 button bit 6 bits selected
at random from box no replacement Find
probability 3 are X bits,
2 are bits and
1 is button bits
45
Hypergeometric Distribution
Probability P(choose x1 from n1, x2 from n2,
etc
46
Hypergeometric Distribution Excel
Example Box contains 6 assays of copper
4 assays of gold choose an essay at
random no replacement 5 trials X number of
copper essays chosen P(X3) Use hypergeometric
47
Hypergeometric Distribution Excel
We are choosing from two categories - copper
(s) and gold (not s) X - number of copper
assays chosen number in s ns 6, number in
not s nns 4 total population nsnns 6 4
10 sample size n 5 hypgeomdist(x,n,ns,n
snns) P(Xltx). P(X3) hypgeomdist(3,5,6,10)
- hypgeomdist(2,5,6,10)
0.476
48
Uniform Distribution
Random variable X uniformly distributed in
altxltb if density functions
1/(b-a) altxltb f(x) 0
otherwise
49
Uniform Distribution Example
Failure rate on bits X f(x) 1/2 2 lt X lt 4
years P (X gt 3) ?34(1/2)dX 0.5X 34
0.54 0.53
0.50 half of bits last more than 3 years P
(2ltXlt 2.5) ?22.5(1/2)dX 0.5X 22.5
0.52.5 0.52
0.25 P(2.7ltXlt3.3) ?
50
Distributions for Econometric Inference

• Y ßo ß1X1 ß2X2 ?
• estimate ßs
• Y bo b1X1 b2X2
• assumptions about distribution of ?
• mean and variance
• gives us distributions of Y, bo, b1, b1
• mean and variance

51
Distributions derived from Normal
2
?21
?
N(0,1)2
?
52
Distributions derived from Normal
2
2
. . . .

?
?2n
N(0,1)2
?
53
?2 Distribution Tests on Variance

• Y (N-1)s2/?2 ?2(N-1) 0 lt Y lt ?
• Want to know
• P(Y lt b)
• P(Ygt a)
• P(altYltb)

54
?2 Distribution Probability from Table
P(Y2 gt 0.103) 0.95 P(Y2 gt 5.991) 0.05 P(Y4 lt
9.488) 1 - P(Y4 gt 9.488) 1 0.05 0.95
P(0.297ltY4lt9.448) ?
55
?2 Distribution Inverse Probability from Table
P(Y3 gt c) 0.95 c 0.352 P(Y4 lt c) 0.99 1
- P(Y4 gt c) 1 0.99
0.01 gt c 13.277
56
?2 Distribution Probability from Excel
P(Y5 gt c) chidist(c,5) P(Y5 gt 2)
chidist(2,5) 0.849 P(Y6 lt c) 1 P(Y6 gt c)
1- chidist(c,6) P(Y6lt4) 1 - P(Y6 gt 4)
1- chidist(4,6) 0.323
57
?2 Distribution Inverse Probability from Excel
P(Y3 gt c) ? gt c chiinv(?,3) P(Y3 gt c)
0.05 gt c chiinv(0.05,3) 7.815 P(Y6 lt c) ?
P(Y6 gt c) 1 - ? gt c chiinv(1- ?,6) P(Y6ltc)
0.1 gt c chiinv(0.90,6) 2.204
58
Distributions for Econometric Inference

• Y ßo ßX1 ß2X2 ? ?Y bo b1X1 b2X2
  assumptions about distribution of ?
• ? mean E(?) 0, variance ?2
• gives us distributions of Y, bo, b1, b2
• Each has mean
• E(Y), E(bo), E(b1), E(b2)
• Each has variance
• ? 2, ? 2bo, ? 2b1, ? 2b2
• ?2 tests on variance
• part of other distributions

59
t-Distribution
N(0.1)
tdf
??2/df
df
60
t-Distribution k degrees of freedom
61
t-Distribution Properties

• Properties
•    When df gt30 approximates Standard Normal
•    Symmetrical with mean 0 and variance
  
• df gt 2

62
t-Distribution Uses

• Used in Econometrics to
• Make inferences on means
• Similar to Normal but tables set up differently
• df bigger the larger the sample
• if n large use normal tables

63
t-Distribution Probabilities Tables
P(t2 gt 1.886) 0.10 P(t4 lt 2.132) 1 - P(t4
gt 2.132) 1 0.05 0.95 P(t2gt-1.886)
P(t2lt1.886) 1- (t2gt1.886) 1 0.10
0.90 P(t4 lt -2.132) P(t4 gt 2.132) 0.05
P(1.638ltt3lt3.182)?
64
t-Distribution Inverse from Tables
P(t3 gt c) 0.025 c3.182 P(t4lt c) 0.90 1 -
P(t4 gt c) 0.90 P(t4 gt c) 0.10 c
1.533 P(t3 gt -c) 0.975 P(t3 lt c) 0.975
1 - P(t3 gt c) 0.975 P(t3 gt c)
0.025 c 3.182 P(t1lt -c) 0.05
P(t1gt c) c 6.314 P(c1ltt3ltc2)
0.90 ?
65
t-Distribution from Excel
P(tdfgt c) ? ? tdist(c,df,1) ?
tdist(c,df,2)/2 P(t10gt 2.10)
tdist(2.10,10,1) 0.031 P(t20lt
1.86) 1 - P(t20gt1.86) 1
- dist(1.86,20,1) 0.961
66
t-Distribution from Excel
P(t15gt-1.56) P(t15lt1.56) 1- (t15gt1.56)
1 tdist(1.56,15,1) 0.93 P(t12 lt
-2.132) P(t12 gt 2.132)
tdist(2.132,12,1) 0.027 P(c1ltt3ltc2) 0.99
?
67
t-Distribution Inverse from Excel
P(tdf gt c) ? ctinv(?2,df1) P(t10 gt c)
0.15 c tinv(0.152,10) 1.093 P(t25lt
c) 0.90 1 - P(t25gt c) 0.90
P(t25gtc) 0.10 c tinv(0.102,25) c
1.316
68
t-Distribution Inverse from Excel
P(t3 gt -c) 0.975 P(t3 lt c) 0.975
1 - P(t3 gt c) P(t3 gt c) 0.025 c
tinv(0.0252,3) c 3.182 P(t1lt -c)
0.05 P(t1gt c) c tinv(0.052,1)6.314 P(c
1ltt17ltc2) 0.90 ?
69
F-Distribution
df1
df2
70
F-Distribution
71
F-Distribution

• Used in Econometrics to 
•    Test between
• two variances
• several means
• subset ?s not simultaneously 0
• Comes from ?2 so 0ltF

72
F-Distribution Relation to Other Distributions


If k2 (denominator df) is large
73
F-Distribution
Properties k1 numerator df k2
denominator df Skewed to right approaches
N(0,1) as k1 and k2 get larger Mean
k2 gt
2 Variance
k2 gt 4  
74
F-Distribution Probabilities from Tables
P(F1,2gt18.513) 0.05 P(F3,6lt4.757) 1 -
P(F3,6gt4.757) 1 0.05
0.95
75
F-Distribution Inverse from Tables
P(F1,2gtc) 0.05gt c 18.513 P(F3,6ltc) 0.95
1 - P(F3,6gtc) P(F3,6gtc) 0.05 gt c 4.757
76
F-Distribution Probabilities from Excel
P(F2,1gt26) fdist(f,df1,dfd2)
fdist(26,2,1) 0.137 P(F6,3lt5.8) 1 -
P(F6,3gt5.8) 1-fdist(5.8,6,3)
1 - 0.0392 0.9608
77
F-Distribution Inverses from Excel
P(F6,8gtc) 0.07 c finv(?,df1,df2)
finv(0.07,6,8) 3.12 P(F9,1ltc) 0.96 1 -
P(F9,1gtc) P(F9,1gtc) 0.04 c finv(0.04,9,1)
376.06
78
Cauchy Distribution

• a
• f(x) p (x2 a2) agt0,
  - ? ltxlt ?
• symmetric around 0
• no moment generating but characteristic function
• like normal but fatter tails
• no variance
• can have bimodal
• t 1 degree of freedom is a Cauchy

79
Exponential

• f(x) ?e-(x/?) xgt0
• excel expondist(x,?,true)
•   applications
• queuing theory
• continuous
• related to Poisson (same lambda)
• Poisson number of repairs
• exponential time between repairs
• life of light bulbs

80
Exponential

graph the exponential function for ? 1, 3, 10
81
Exponential Example

• Example When drilling oil well
• breakage or lost tool down hole
• fishing for it
• expensive
• no luck fishing
• deviate well
• Suppose fish time t exponential ?1/3.
• longer than 7 hours better to deviate
• percent of time better to deviate?
•   1-expondist(7,1/3,true)0.096972

82
Integrate Exponential Functions

• Make review mineral examples to show how to
  integrate
• ex, e-x, eax

83
Integrate Exponential Functions Rulesadd example

• General integration rule
• ? ekx dx ekx / k for example ? ex
  dx ex
• Integration by substitution rule
• ? x ex2 dx
• Substitute u x2 du 2x dx or dx du/2x
• Then
• ? x eu (du/2x) ½ ? eu du ½ eu

84
Integrate Exponential Functions Rules

• Integrate by parts
• ? x ex dx
• Let f (x) x and g\ ex, then f\(x) 1 and g(x)
  ex
• ? x ex dx f (x) . g (x) - ? g (x) . f\ (x)
  dx
• x ex - ? ex dx x ex ex
• To check take the derivative for this expression
• (x ex - ex) it will give (x ex )

85
Integrate Exponential Functions
4-85
integrate by parts v x u e-?x dv
dx du -?e-?xdx
86
4-86
Integrate Exponential Functions
87
Integrate Exponential Functions
4-87

• If ? 1 find P(x lt 3)

88
Log Normal
X is lognormally distributed if Y Ln(X) is
normally distributed Uses -
failure rates - mineral deposits

89
Log Normal

m mean s sigma standard deviation

Sourcehttp//mathworld.wolfram.com/LogNormalDistr
ibution.html
90
Log Normal

91
Log Normal in Excel
lognormal distribution is
lognormdist(x, mean, std. dev.) lognormdist(
1.5,0.5,0.5) 0.425019

92
Chapter 4 Sum Up

• Special Distributions Powerful tools
• Binomial Distribution
• Normal Distribution
• Poisson Distribution
• Relations Between Distributions
• Binomial and Normal
• Binomial and Poisson
• Poisson and Normal

93
Chapter 4 Sum Up

• Central Limit Theorem
• Multinomial Distribution
• Hypergeometric Distribution
• Uniform Distribution
• Distributions from Normal
• ?2 Distribution
• t-Distribution
• F-Distribution
• Y ßo ß1X1 ß2X2 ?

94
Chapter 4 Sum Up

• Cauchy
• Exponential
• Lognormal

95
End of Chapter 4