Probability Density Functions Jake Blanchard Spring 2010 Uncertainty

Probability Density Functions Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers 1

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 Engineers 2

Random Variables �Let y 1 and y 2 represent the values of the two dice �Let x=y 1+y 2 �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 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 5 4 3 2 1 Uncertainty Analysis for Engineers 3

Probability Distribution Function �We can calculate a probability from this table and plot the probability against the sum 0, 18 Probability of Occurrence 0, 16 0, 14 0, 12 0, 1 0, 08 0, 06 0, 04 0, 02 0 2 3 4 5 6 7 8 Sum of Two Dice 9 10 11 12 Uncertainty Analysis for Engineers 4

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 Engineers 5

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 Engineers 6

Cumulative Probability 1 0, 9 0, 8 Cumulative Probability 0, 7 0, 6 0, 5 0, 4 0, 3 0, 2 0, 1 0 1 2 3 4 5 6 7 Sum of Two Dice 8 9 10 11 Uncertainty Analysis for Engineers 7

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 or Uncertainty Analysis for Engineers 8

Continuous Functions �Consider the decay of a radioactive particle �The probability it will survive beyond time ti is Pr(t>ti)=exp(- ti) �Hence, the CDF is given by Pr(t<=ti)=F(ti)=1 -exp(- ti) �This is plotted for =1/s on the next slide Uncertainty Analysis for Engineers 9

CDF for radioactive decay 1 0, 9 0, 8 F(time) 0, 7 0, 6 0, 5 0, 4 0, 3 0, 2 0, 1 0 0 0, 5 1 1, 5 2 2, 5 3 3, 5 4 4, 5 time Uncertainty Analysis for Engineers 10

Decay Example =0. 1, the probability that a particle will decay between 4 and 5 seconds is given by P(4<t<=5)=F(5)F(4)=[1 -exp(-0. 5)]-[1 -exp(-0. 4)]=0. 063 �For Uncertainty Analysis for Engineers 11

Characterizing Distributions Functions �We will see later how to characterize these functions using ◦ ◦ ◦ Mean Median Standard Deviation Skewness Kurtosis Etc. Uncertainty Analysis for Engineers 12

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 Engineers 13

Joint PDFs �Two continuous random variables are said to have a bivariate or joint pdf f(x, y) if Uncertainty Analysis for Engineers 14

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 Engineers 15