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Unformatted text preview: Space IOE 202: lecture 10 outline Announcements Last time... Modeling continuous random quantities: density functions Uniform, Normal and Exponential random variables IOE 202: Operations Modeling, Fall 2009 Page 1 Space Last time: incorporating uncertainty into models and decision making Replacing uncertain, or random, quantities by their expected, or average, values may lead to misleading estimates of outcomes, and hence poorly informed decisions Including uncertainty into models for decision-making Characterizing random quantities taking on a discrete set of values
New product introduction Walton bookstore Incorporating uncertainty into modeling and analysis via simulation
Ordering calendars at Walton bookstore Setting the mean ﬁll rate at Bottleco IOE 202: Operations Modeling, Fall 2009 Page 2 Space Describing uncertainty
How do we characterize uncertainty about the value of a parameter to incorporate it into a model? For uncertain quantities that take on values in a discrete set, can specify the probability (likelihood, chance) of the quantity taking on a particular value, for each possible value.
Probabilities must be nonnegative Probabilities of all possible outcomes must add up to 1 Will you have to stop at the traﬃc light at Geddes driving down Huron Parkway? What number will a rolled die display? What about the sum of numbers on two dice? How many pairs of shoes will the store sell tomorrow? Characterizing discrete random quantities and events:
Describing random quantities that take on values in a continuous set is diﬀerent:
If you spin the wheel at the “Wheel of fortune,” what chance is there that the spinner ends up at exactly the angle of 5o ? What is the probability that the result is “Bankrupcy”?
Page 3 IOE 202: Operations Modeling, Fall 2009 Space Continuous (or “divisible”) random variables (r.v.) Examples: At what angle does the end of the spinner at “Wheel of Fortune” end up? A random variable; can take on any value between 0 and 360. How much time will elapse until an observed radioactive particle decays? A random variable; can take on any nonnegative value. What is the height of a randomly selected adult woman in the US? A random variable; can take on any nonnegative value, although realistically will not see values below or above some limits. A continuous random variable has probability 0 of taking on any particular value.1 To characterize the outcomes of such random variables, instead, we describe the probabilities that the value will fall in ranges of interest.
This is because the probability that the variable takes on any particular value cannot be found by adding the probabilities for individual values.
IOE 202: Operations Modeling, Fall 2009 Page 4 1 Space Density functions of continuous r.v.’s: Intuition Suppose we spin a “wheel of fortune” which is divided into 24 diﬀerent segments.
The probability that the needle lands in any particular segment 1 is 24 A larger number of them will have her height between 5’4” and 5’5” than between 6’4” and 6’5”. So, possible outcomes of the random variable in question are more densely concentrated over the interval 5’4” to 5’5” than the interval 6’4” to 6’5”. Suppose we selected a sample of adult women at random
A density function of a random variable is a geometric representation of the above intuition. IOE 202: Operations Modeling, Fall 2009 Page 5 Space Probability density function of a Uniform r.v. Probability that the angle is in a particular range is the area under the curve across that range. Example: Probability density function of a random variable uniformly distributed between 0 and 360: To plot a histogram: prepare a range of categories, then select “Tools → Data Analysis → Histogram”; select appropriate ranges and select “Chart output”
IOE 202: Operations Modeling, Fall 2009 Page 6 Space Density function of a Normal r.v. Deﬁnition: Random variable X has a Normal distribution with mean µ and standard deviation σ if its density function is √
1 x −µ 2 1 e − 2 ( σ ) for − ∞ < x < ∞ 2πσ Probability density functions of three diﬀerent random variables, all with normal distributions:
0.14 0.12 N(64,3) N(60,3) N(64,5) 0.1 0.08 0.06 0.04 0.02 0 50 55 60 65 70 75 80 IOE 202: Operations Modeling, Fall 2009 Page 7 Space Observations about normal distributions
Area under each curve is 1! Symmetric around their means, i.e., a normal random variable is as likely to be ∆ units below its mean as ∆ units above, for any ∆ Things in common: if X is a Normal r.v., the probability that it will take on value within k standard deviations of its mean depends only on k :
Prob(µ − σ ≤ X ≤ µ + σ ) = 0.6826 Prob(µ − 2σ ≤ X ≤ µ + 2σ ) = 0.9544 Prob(µ − 3σ ≤ X ≤ µ + 3σ ) ≈ 1 Prob(µ − 1.96σ ≤ X ≤ µ + 1.96σ ) = 0.95 Prob(X ≤ µ + σ ) = 0.8413 Prob(X ≥ µ − 2σ ) = 0.97720, etc. IOE 202: Operations Modeling, Fall 2009 Page 8 Space Sampling from Normal distirbution To sample values from Normal distribution with mean µ and standard deviation σ , we can:
Option 1 Use Random Number Generation to generate the desired number of replications of Normal with mean µ and standard deviation σ Option 2 Use Random Number Generation to generate the desired number of replications of Normal with mean 0 and standard deviation 1.2 Then multiply each number by σ and add µ (in that order).
Useful for performing analysis (BottleCo problem; Homework, problem 3) These represent the deviations of the numbers from the mean, counted in the number of standard deviations.
IOE 202: Operations Modeling, Fall 2009 Page 9 2 Space Importance of the Normal distribution Many important unknown quantities that we need to model in applications can be realistically modeled as being Normal random variables Simple to analyze To decide whether this is a realistic model, check the frequency of the above deviations from the mean (other values can be obtained from the standard normal table) IOE 202: Operations Modeling, Fall 2009 Page 10 Space Density function of an Exponential r.v. Deﬁnition: Random variable X has an exponential distribution with rate λ > 0 if its density function is f (x ) = λe −λx for x > 0
5 4.5 Exponential with lambda=3 Exponential with lambda=5 Exponential with lambda=1 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Observation: the mean of an exponential random variable with rate λ is 1/λ (and so is the standard deviation!)
IOE 202: Operations Modeling, Fall 2009 Page 11 Space The memoryless property of exponentials Example: suppose that the lifetime (in years) of ShineBright lightbulbs is an exponentially distributed random variable, with rate λ = 0.5. Let us randomly select a ShineBright lightbulb. 1. What is its expected lifetime (i.e., what is the average lifetime of such lightbulbs)? 2. What is the probability that it will fail in less than a year? 3. What is the probability that it will last at least 1.5 years? 4. Suppose the lightbulb has been on for a year, and has not failed so far. What is the probability it will last at least another 1.5 years? The ﬁrst three questions can be answered analytically, but let us try to answer them with a simulation... Observe that the exponential distribution is memoryless: answers to the second and fourth questions are the same!
IOE 202: Operations Modeling, Fall 2009 Page 12 ...
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- Fall '09