1. Project Management for Software Engineers (Summer 2017)
LECTURE 12
Risk Quantification Basics
July 19, 2016
(9:00 am – 11:40 pm PST)
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018

2. Risk = Uncertainty Risk quantification means:
“How much” uncertain are we about a particular outcome
Supplier  Input  Process  Output  Customer: Focusing on the I.P.O:
If we know “how much” uncertain we are about the Inputs, we can agglomerate all uncertainties to calculate the uncertainty about the output, using statistical techniques (simulation)
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018

3. A Crash Course on Probabilities
A + Ac = 1
P(A or B) = P(A) + P(B) – P (A and B)
Mutually Exclusive events: P(A and B) =0
Conditional Probability
P(A|B) = P(A and B) / P(B)  P(A and B) = P(A|B)xP(B)
Joint Probabilities: Decision Tree  (pp )
Independent Events: P(A|B)=P(A)  P(A and B) =P(A)xP(B)
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018

4. A Crash Course on Statistics
Descriptive Statistics
Current Data attributes (central tendency & dispersion)
Histograms: Location, Spread, Shape
Inferential Statistics (Enumerative Studies)
Sample  Population (Hypothesis Testing)
Key: Proper Sampling  Representative Sample
Predictive Statistics (Analytic Studies)
DOE, Regression, ANOVA
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018

5. Descriptive Statistics
Central tendency: Mean, median, mode
Dispersion: Variance, Min, Max, Range
Shape: Histogram, Skewness, Kurtosis
Others: Count, Sum
Excel Example:
Tools  Data Analysis  Descriptive Statistics
Tools  Data Analysis  Histogram
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018

6. Statistical Distributions
Discrete: Binomial, Hypergeometric, Poisson, …
Continuous: Normal, Lognormal, Exponential, …
Interrelationships
Binomial  Poisson
Normal  Lognormal
Poisson  Exponential
Sources of abnormality:
Recording & Measurement errors, Multiplicity, truncation, natural skewness, tampering (Deming experiments)
Understand data & sources of variation before sampling
Bi (p<0.5 & n>20) == Poisson (np)
Bi: Flipping a coin
Poisson: # of cars entering an intersection
Exponential: Interarrival of of cars entering an intersection
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018

7. Inferential Statistics
Sampling: Select an “unbiased” subset of an entire group
Simple Random Sampling (SRS): For homogeneous sets
Stratified sampling: Group by some attributes (internally homogeneous)
Clustering: Similar groups, different components (“Naturally” clustered, internally heterogeneous) – Less Precise & More Economic
Systematic: Every (N/n)th sample, or some other rules (Easy, but prone to bias)
Examples:
Random, NOT arbitrary!
Stratified: Sampling City population, in age groups
Stratified: Exponential Distribution
Clustered: Sampling random pallets of packed products
Systematic: Sampling every 8th house of a street with 120 houses
NOT REQUIRED
Stratified: Exponential Distribution tail has bigger weight
Systematic could be biased

8. Representative Samples
If the sampling is performed properly, the sample “represents” the population
Parameter Sample Population
Size n N
Mean x̄ m
St. Dev. s s
CLT – For very large population (n/N<0.05) and “Truly” representative samples:
E(x̄) = m & s(x̄)=s /√n  x̄ ~ N ( m , s /√n )
Example: P. 280, Examples 6.13 & 6.14
Note: N(x, m, s) == Ns[(x-m)/s, 0, 1]  Ns Table (pp.A3&A4)
In CLT: x̄ ~ Ns [(x̄- m)/(s /√n ),0,1]
NOT REQUIRED
University of Southern California, Industrial & Systems Engineering
9/18/2018

9. From “Point” To “Range” Estimate
Confidence Level (CL) = 1 - a
a = Probability of a parameter falling outside a given range. (e.g. probability of defect)
Example: “90% CI for m is x̄ + q” means “In 100 samples, for 90 of them: the range [x̄ - q x̄ + q ] contains the true value of m”
Equations: P. 281
Examples 6.15 & 6.16 (p. 282)
NOT REQUIRED
University of Southern California, Industrial & Systems Engineering
9/18/2018

10. Estimating reality by experimentation
What is simulation?
Problem: Flipping a coin as many times as we get 3 heads; Cost: $1 per coin-flip, reward: $8 for 3 heads
Can you develop a formula to get the answer?
Yes you can, but it’s a tedious and complicated process.
Can it be done easier? Just walk the talk:
Generate a random integer (0, 1) Excel: Int((Rand()+0.5))
Repeat until you get three 1’s (heads)
If we repeat this experiment many many times, we can develop a statistical distribution for the sample size (i.e. number of flips that returns 3 heads)
Example calculations in Excel (handout)
Estimating reality by experimentation
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018

11. Monte-Carlo Simulation
Process: If Y=f(X1,X2,X3,…) and statistical distribution of Xi’s are known as F(Xi), then:
U = F(Xi)  Xi = F-1(U), where U is a probability, a random number between 0 and 1 (100%)
With numerous iterations, statistical distribution of Y and its statistical characteristics can be modeled and the probability of Y<=Y0 can be estimated with a very high accuracy
The more iterations, the more accurate the statistical distribution for Y
For each Xi:
For the entire model (Y):
N=500
N=1000
N=100
N=200
Source: Queueing Methods, Randolph W. Hall (1991), P. 59
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018

12. A Basic Example A Typical customer service: Poisson Process Trial #5
𝐹 𝑥 =1− 𝑒 −l𝑥 , SET: R[0−1]=𝐹 𝑥  𝑥= ln⁡ 𝑅 −l
Assume l=4 customers / hour based on historical data
How many customers will arrive in 2 hours?
Trial #5
N=5 #5
N=4 #4
Trial #4
N=2 #3
Trial #3
Trial #1
N=6 #1
N=5 #2
Trial #2
Also see Example: Hall. P. 185
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018

13. Simulating PM Metrics NPV : pp 310-318 CPM/PERT Scheduling: pp 367-372
University of Southern California, IMSC/SSU CERTIFICATION PROGRAM
9/18/2018