Chapter 11: Selected Quantitative Relationships (pt. 1) ISE 443 / ETM 543 Fall 2013

The development of large, complex systems involves the use of quantitative relationships 2 443/543 – 11 To predict system performance before it is built  response times  success rates  system availability  system reliability  error rates  catastrophic failure Because of the uncertainty involved, probability is the most used quantitative relationship in systems engineering (and, to a lesser extent, project management)

To start, we’ll review the fundamentals Basic probability – Ned and Ryan S. Discrete distributions  binomial distribution – Lexie and Rebecca  poisson distribution – Filipe, Xin, and Yuan Continuous distributions  normal distribution – Alfred and James  uniform distribution – Justin and Gleidson  exponential distribution – Elvire and Ryan K. Means and variances – Jamie and Geneve Sums of variables – Isis and Jessica Functions of random variables – Charmaine and Melanie Two variables (joint probability) – Celso and David Correlation – Lucas and Kyle 3 443/543 – 11

Your turn … Form into pairs. As a pair review the list on the previous slide and select a topic you wish to review for the class.  You may want to have at least one “backup” topic. Once you have decided on a topic, STAND UP. When you are recognized (I will go in order) state your selected topic.  If someone else selects your topic, move to your “backup”. If you don’t have a backup, sit down until you have selected another. Review the section in the book related to your topic and develop a brief summary to share with the class today (you will have 10 minutes to prepare).  10 minutes after the last topic is selected, we will go through each topic as a class and discuss what you have so far.  Make note of the feedback you receive. You may find this useful when completing the homework assignment /543 – 11

Basic probability For any given event P > 0 For a certain event, P = 1, if it can’t happen P=0, if it may happen it’s between 0 and 1. If 2 events are mutually exclusive, add the probabilities to find the probability of either one happening To find probability of BOTH happening, multiply the 2 probabilities Other rules for independence, non-independent, etc /543 – 11

Discrete distributions: Binomial Arises when there are repeated independent trials with only 2 possible results If P(success) = p and P(failure) = q, then p+q=1 The distribution is the equation on page 345. The distribution defines the probability of exactly x successes in n independent trials Example at top of page /543 – 11

Discrete distributions: Poisson Deals with the issue of software reliability May be used in situations for which events happen at some rate and we wish to ascertain the probability Example on page 346 Formula is also on /543 – 11

Continuous distributions: Normal AKA Gaussian Bell-shaped curve Density function on page 347, eq Integrate (p(x < X)) OR use tables (p(z <Z)) 8 443/543 – 11

Continuous distributions: Uniform Occurs when there is a constant probability over a range Cumulative density function is a triangle from 0 to 1. Equations for the mean and variance are on page 350 Example on page 350 (arrow shot at bullseye) 9 443/543 – 11

Continuous distributions: Exponential Density and cumulative dist function are illustrated on pg 350 The CDF starts at 0 and approaches 1 asymptotically Formulas are given on pg 350 Widely used in reliability theory wherein the value of lambda is taken to be a constant failure rate Failure rate and MTBF are reciprocals of each other /543 – 11

Means and variances Mean equation is on pg 340 (both discrete and continuous) Example of rolling dice Variance eq on pg 341 Variance indicates the spread of the distribution Std deviation is the square root of the variance A critical performance measure is the S/N ratio, which is … /543 – 11

Sums of variables Probabiltiy of the sum = sum of the probabilities (when the variables are mutually exclusive) Eq on pg 341 Distribution of the sum is the convolution of the individual distributions Variance of the sum is the sum of the variances only when the individual distributions are independent The mean value of the sum is the sum of the mean values /543 – 11

Functions of random variables Arise when we make a functional transformationa dn wish to examing the result Value is to track how a varaible behaves as it is processed through a system Example on pg /543 – 11

Two variables (joint probability) Explains the joint behavior of 2 independent varialbes where the prob of both happening can be exrpessed by P(x,y)= P(x)P(y) Example on pg 344 Mean or expected value eq on the same page /543 – 11

Correlation 2 variables have a relationship Correlation coefficient eq on pg 345 (11.22) Normalized to values between -1 and perfect positive correlation, 0 is no correlation, -1 is perfect neg correlation Found /543 – 11