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Probability Density Functions Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers1

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Random Variables We will spend the rest of the semester dealing with random variables A random variable is a function defined on a particular sample space For example, if we roll two dice there are 36 possible outcomes – this is the sample space The sum of the two dice is the random variable Uncertainty Analysis for Engineers2

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Random Variables Let y1 and y2 represent the values of the two dice Let x=y1+y2 x can take on any one of 11 values between 2 and 12, with some more common than others The relative likelihood of rolling each of the possible sums is Uncertainty Analysis for Engineers3 23456789101112 12345654321

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Probability Distribution Function We can calculate a probability from this table and plot the probability against the sum Uncertainty Analysis for Engineers4

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Continuous Probability Distribution Functions Define the pdf [f(x)]such that the probability that x falls between a and b is given by Uncertainty Analysis for Engineers5

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Cumulative Probability What if we are interested in the probability that the sum is at or below some value For example, the probability that the sum is less than or equal to 4 is 6/36=1/6=0.167 We can plot this value as a function of the sum Uncertainty Analysis for Engineers6

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Cumulative Probability Uncertainty Analysis for Engineers7

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Cumulative Probability We call this the cumulative distribution function (CDF) It has a minimum of 0, a maximum of 1, and is monotonic For the example of the sum of two dice, the CDF is Uncertainty Analysis for Engineers8 or

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Continuous Functions Consider the decay of a radioactive particle The probability it will survive beyond time t i is Pr(t>t i )=exp(- t i ) Hence, the CDF is given by Pr(t<=t i )=F(t i )=1-exp(- t i ) This is plotted for =1/s on the next slide Uncertainty Analysis for Engineers9

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CDF for radioactive decay Uncertainty Analysis for Engineers10

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Decay Example For =0.1, the probability that a particle will decay between 4 and 5 seconds is given by P(4

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Characterizing Distributions Functions We will see later how to characterize these functions using ◦ Mean ◦ Median ◦ Standard Deviation ◦ Skewness ◦ Kurtosis ◦ Etc. Uncertainty Analysis for Engineers12

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Bivariate Distributions Sometimes we work with more than one random variable. These can be correlated, so it is appropriate to define a single pdf that governs both variables simultaneously We call this a joint probability density function Uncertainty Analysis for Engineers13

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Joint PDFs Two continuous random variables are said to have a bivariate or joint pdf f(x,y) if Uncertainty Analysis for Engineers14

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Types of pdfs We have many choices for functional forms of pdfs Our goal is to represent reality Ultimately, we need data to validate our choice of pdf We’ll discuss this later Next, we’ll look at some of the common forms Uncertainty Analysis for Engineers15

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Exercises (1) 1. In the usual dice experiment, let A, B and C, be the following events: A = {1,7,6} and B = {1,2,7,5} and C = {3,5} Find the following:

Exercises (1) 1. In the usual dice experiment, let A, B and C, be the following events: A = {1,7,6} and B = {1,2,7,5} and C = {3,5} Find the following:

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