Probability Density Functions

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

2. 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

3. 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 Engineers

4. Probability Distribution Function • We can calculate a probability from this table and plot the probability against the sum Uncertainty Analysis for Engineers

5. 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

6. 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

7. Cumulative Probability Uncertainty Analysis for Engineers

8. 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

9. 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

10. CDF for radioactive decay Uncertainty Analysis for Engineers

11. Decay Example • For =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 Uncertainty Analysis for Engineers

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

13. 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

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

15. 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