Download Data, Statistics, Probability 3 : Probability

Data, Statistics, Probability 3 :
Probability distributions
Christopher Grigoriou
1
Executive MBA - HEC Lausanne
2007/2008
Probability Distributions
2
Discrete distributions:
Binomial
Uniform
Continuous distributions:
Uniform
Exponential
Normal
t-distribution
Confidence intervals for the Normal distribution
Normal approximation to the Binomial distribution
Executive MBA - HEC Lausanne
2007/2008
Binomial distribution
E.g. Customer enquiries
Each customer: makes a booking with probability p
Hypothesis: Behaviour of different customers independent
Consider the total number of calls per day (n)
Random variable: X = number of these n customers who make a booking
==> Binomial distribution with parameters n and p
X ~ B(n,p), E(X) = np, Var(X) = np(1-p)
Type of questions:
Probability that more than half the customers make a booking
Probability that at least x bookings
3
Executive MBA - HEC Lausanne
2007/2008
Example: A probability tree approach
4
Number of enquiries = 3 (n)
Probability that an enquiry results in a booking = 0.4 (p)
Probability that these 3 calls result in exactly 2 (r) bookings
= Pr(r = 2 | n = 3, p = 0.4)
=
Probability that these 3 calls result in at least 2 bookings
= Pr(r ≥ 2 | n = 3, p = 0.4)
=
Executive MBA - HEC Lausanne
2007/2008
Distribution of Proportions
E.g. Customer enquiries
Each customer: makes a booking with probability p
Hypothesis: Behaviour of different customers independent
Consider the total number of calls per day (n)
Random variable: X = number of these n customers who make a booking
==> Binomial distribution with parameters n and p
X ~ B(n,p), E(X) = np, Var(X) = np(1-p)
Random variable: Y = X/n = Proportion of customers who make a booking
E(Y) = E(X/n) = (1/n) E(X) = p
Var(Y) = Var(X/n) = (1/n2) Var(X) = p(1-p)/n
5
Executive MBA - HEC Lausanne
2007/2008
Normal Distribution
Probability density
Example: Estimate the $ - CHF exchange
rate for next June
==> Random variable X
E.g. E(X) = 1.6, SD(X) = 0.2
68%
2.5%
2.5%
Assume: Normal distribution
Values close to the mean are most likely
==> Small probability of extreme value
1
1.2
1.4
E(X) - 2SD
6
1.6
E(X)
1.8
2.0
2.2
E(X) + 2SD
Executive MBA - HEC Lausanne
2007/2008
Shape and location
N(1.6, 0.2)
N(1.6, 0.05)
1
1.2
1.4
1.6
1.8
2
2.2
N(0, 0.2)
7
-0.6
-0.4
-0.2
N(1.6, 0.2)
0
0.2
0.4
0.6
1
1.2 Executive
1.4 MBA
1.6 - HEC
1.8 Lausanne
2
2.2
2007/2008
Asymmetric distributions
Distributions: X1, X2
X1
X2
Var(X1) = Var(X2)
E(X1) = E(X2)
8
Assume these are profit
distributions
Which do you prefer?
E(X1) = E(X2)
Executive MBA - HEC Lausanne
2007/2008
Relating the Density function to the Cumulative
distribution
1,00
0,75
0,50
68%
2.5%
1
2.5%
1.2
1.4
1.6
1.8
2.0
2.2
0,25
0,00
0
9
0,5
1
1,5
2
Executive MBA - HEC Lausanne
2007/2008
2,5
Standard Normal Distribution N(0,1)
1.2
1.4
1.6
1.8
2.0
2.2
X ~ N(1.6, 0.2)
-0.6 -0.4
-0.2
0
0.2
0.4
0.6
X - E(X) ~ N (0, 0.2)
-1
0
1
2
3
1
-3
10
-2
X - E(X) ~ N(0,1)
SD(X)
= z-score
Executive
MBA - HEC Lausanne
2007/2008
Some critical values of N(0,1)
Pr(Z > z)
0
11
z
Pr(Z>z)
z
50%
0
45%
0.13
40%
0.25
35%
0.39
30%
0.52
25%
0.67
20%
0.84
15.90%
1
10%
1.28
5%
1.64
(round to 2)
2.50%
1.96
2.30%
2
1%
2.33
0.62%
2.5
0.50%
2.57
0.13%
3
0.10%
3.09
Executive MBA - HEC Lausanne
2007/2008
Confidence intervals
Example:
95% Confidence interval:
90% Confidence interval:
12
X = $ - CHF exchange rate for next June
X ~ N(µ = 1.6, σ = 0.2)
Executive MBA - HEC Lausanne
2007/2008
t-distribution
t-score
0.0
Degrees of freedom
1
5
10
20
30
50
100
0.500 0.500 0.500 0.500 0.500 0.500 0.500
1.0
0.250 0.182 0.170 0.165 0.163 0.161 0.160
2.0
0.148 0.051 0.037 0.030 0.027 0.025 0.024
2.5
0.121 0.027 0.016 0.011 0.009 0.008 0.007
3.0
0.102 0.015 0.007 0.004 0.003 0.002 0.002
Tail
probability
13
Similar to Standard normal distribution
Family of distribution with one parameter:
degrees of freedom n
As n goes to infinity, t(n) converges to N(0,1)
N(0,1)
Degrees of freedom
1
5
10
20
30
50
100
50.0%
0.00
0.00
0.00
0.00
0.00
0.00
0.00
10.0%
3.08
1.48
1.37
1.33
1.31
1.30
1.29
5.0%
2.5%
1.0%
6.31
12.71
31.82
2.02
2.57
3.36
1.81
2.23
2.76
1.72
2.09
2.53
1.70
2.04
2.46
1.68
2.01
2.40
1.66
1.98
2.36
t(2) t(10)
-3
-2
-1
0
1
2
3
Executive MBA - HEC Lausanne
2007/2008
Normal approximation to the binomial
distribution
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X = Number of successes
Assume X ∼ B(n,p)
If np > 5 and n(1-p) > 5 and 0.1 < p < 0.9
Then can approximate X by N(µ =np, σ2 = np(1-p))
X/n = Proportion of successes
If X ∼ N(µ =np, σ2 = np(1-p))
Then X/n ∼ N(µ =p, σ2 = p(1-p)/n )
Executive MBA - HEC Lausanne
2007/2008