IERG 3050: Review of Probability and Statistics Week 2ierg3050/Lectures/week2-lecture.pdf · •Model a probabilistic system •Choose the input probability distributions •Generate

IERG 3050: Review of Probability and Statistics

Week 2

Bolei ZhouDepartment of Information Engineering

Announcement

This Week’s Tutorial (temporarily rescheduled): • Thursday: 15:30 pm – 16:15 pm at SHB 801• Friday: 14:00 pm – 14:45 pm at SHB 801

Course project proposal due: Sept.22, 2019

Motivation• Simulation is more than flowcharts and programming!

• One needs to apply probability and statistics in various stages:

• Model a probabilistic system• Choose the input probability distributions• Generate random numbers from given distributions• Perform statistical analyses of the simulation output data• Validate the simulation models

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Outline

• A comment on Arrival Processes and Service Times• Probability Models, Conditioning, and Independence• Random Variables: Discrete vs. Continuous• Cumulative Distribution Function• Joint Probability Distribution• Mean and Variance• Covariance and Correlation

• Reading: Chapter 4

□ Acknowledgement: Prof. Minghua Chen, Rosana Chan, Prof. Angela Zhang, Prof. Jianwei Huang, and Prof. Pascal Vontobel for contributing to the slides

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Queueing Systems

• Single-queue-single-server• Multiple-queue-multiple-server

A Comment on Arrival Processes and Service Times

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Outline



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Probability Model

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Event

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Probability

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Conditional Probability

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Independence

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Exercises

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Exercises (Wall street bank interview question)

“Let’s play Russian Roulette. Here’s a gun, a revolver. Here’s the barrel of the gun, six chambers, all empty. Now watch me as I put two bullets into the barrel, into two adjacent chambers. I close the barrel and spin it. I put a gun to your head and pull the trigger. Click. Lucky you! Now I’m going to pull the trigger one more time. Which would you prefer: that I spin the barrel first or that I just pull the trigger?”

• A variant: How about that the two bullets can be in any positions.

Exercises (Wall street bank interview question)

IERG 3050: Review of Probability and Statistics

Week 2 Lecture 5

Bolei ZhouDepartment of Information Engineering

Outline



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Comment for Last Lecture

P(A) = 0.30 + 0.10 + 0.12 = 0.52the conditional probability：P(A|B1) = 1,P(A|B2) = 0.12 ÷ (0.12 + 0.04) = 0.75,and P(A|B3) = 0.

Comment for Last Lecture

“Let’s play Russian Roulette. Here’s a gun, a revolver. Here’s the barrel of the gun, six chambers, all empty. Now watch me as I put two bullets into the barrel, into two adjacent chambers. I close the barrel and spin it. I put a gun to your head and pull the trigger. Click. Lucky you! Now I’m going to pull the trigger one more time. Which would you prefer: that I spin the barrel first or that I just pull the trigger?”

Random Variable

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Random Variable

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Discrete Random Variable

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Continuous Random Variable

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Continuous Random Variable

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Outline



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Cumulative Distribution Function

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Cumulative Distribution Function

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Outline



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Joint PMF and Conditional PMF

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Joint PDF and Conditional PDF

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Independence of Random Variables

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Independence of Random Variables

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Exercise

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Outline



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Mean (Expected Value)

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Mean (Expected Value)

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PMF

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Expectation

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Variance

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Variance

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Properties of Expectation

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Example

• Suppose that two factories supply light bulbs to the market. Factory X's bulbs work for an average of 5000 hours, whereas factory Y's bulbs work for an average of 4000 hours. It is known that factory X supplies 60% of the total bulbs available. What is the expected length of time that a purchased bulb will work for?

• Applying the law of total expectation, we have: E(L) = E(L|X)P(X) + E(L|Y)P(Y) = 5000*0.6 + 4000*0.4 = 4600

Properties of Variance

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Discrete Random Variables

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Bernoulli Distribution

Flip a coin, the chance to get a headDiscrete probability distribution: Pr(X=1) = p, Pr(X=0) = 1-p• E(X) = p• Var[X] = p(1-p), why?• E[X2] = Pr(X=1) * 12 + Pr(X=0) * 02 = p• Var[X] = E[X2] – E[X]2=p-p2 = p(1-p)

Binomial Distribution

• Flip a coin n times, numbers of head we can get.

• Relation to Bernoulli distribution?

• Practice: compute the mean and variance of binomial distribution

Example for binomial distribution

Suppose a biased coin comes up heads with probability 0.3 when tossed. What is the probability of achieving 0, 1,..., 6 heads after six tosses?

Geometric Random Variable

• Flip a coin, until you get a head• Distribution over the number of trials needed to get the first success

in repeated Bernoulli trials.


• Memoryless property: P(X > x+y | X > x) = P(X > x)• Example: Pr(X > 40 | X > 30) = Pr(X>10)• Proof:


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• Mean of the geometric random variable E(X) = 1/p

Coupon collector’s problem

• How many snacks you have to buy to collect all n different types of coupon?

ti has geometric distribution with expectation 1/pi

Exponential Random Variable

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Gaussian Random Variable

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Outline



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Covariance

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Correlation

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Summary of Concepts and Principles

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Course Project

Group-forming and proposal due on Sept 22• Reply to a dedicated thread created by Yinghao (TA)

with your project title and group member in the reply, and project proposal attached

• Sample proposal on piazza• Open-ended: as long as you use simulation and

statistical analysis to analyze/design (abstract versions of) real-world systems, it will be fine

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Course Project Steps (Suggestions only)

• Collection of data;• Modelling and validation of random variables;• Modelling of the system;• Implementing a simulation program of the system model;• Reporting the results;• Stating some conclusions, comments, and/or suggestions for

improvement.

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Thank you!

Documents

IERG 3050: Review of Probability and Statistics Week 2ierg3050/Lectures/week2-lecture.pdf · •Model a probabilistic system •Choose the input probability distributions •Generate