Probability. Example 1 9/23/ Probability Density Functions. 4-1 Continuous Random Variables

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1 4-1 Continuous Random Variables Examples: Length of copper wire Changes in ambient temperatures Thermometer reading in degrees for freezing water Time between detections of a particle with a Geiger counter Loading on a long thin beam 4-2 Probability Density Functions f(x) Figure 4-1 Density function of a loading on a long, thin beam. b a f ( x) dx Probability 4-2 Probability Distributions and Probability Density Functions Probability determined from the area under f(x). Major difference from Discrete r.v.: No mass probability at single value. 4-3 Cumulative Distribution Functions Continuous Uniform Distribution Example 1 F(x) Continuous Uniform pdf 1

2 Example Mean and Variance 4-4 Example 1 (cont d) Major difference: Sum versus integral. P115: Read the following examples from the textbook: 4-5 CDF of continuous uniform dist n Example 4-6 Example 4-7 Example 4-8 a b 2

3 4-6 Normal Distribution (see illustration athttp://vimeo.com/ ) Homework from 6 th edition: Exercise 4-2: 4-7, 4-8, 4-11, 4-15 Exercise 4-3: 4-17, 4-23, 4-29, 4-33 Exercise 4-4: 4-35, 4-37, 4-45, 4-49 Exercise 4-5: 4-51, 4-53, : Odd problems Notation: X~N(, 2 ) Watch another video at Normal density function (bell shaped curve) 4-6 Rule of Thumb 4-6 Standard Normal Distribution 4-6 Standardized Score or Z-score Notation: Z ~ N(0,1) 3

4 Table III Table III 4-6 Standard Normal Table III Let Z~N(0,1). Find P(Z<1.5)=? P(Z<1.53)=? Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads (at the freezing point of water) between 2.00 and 1.50 degrees. III III (1) P (Z < 2.00) = (2) P (Z < 1.50) = (3) P ( 2.00 < Z < 1.50) = = Example 4-13 Let X=the current in milliamperes. Then X~N(10, 4). X Z ~ N(0,1) and z P(X>13)=P(Z>1.5)=1-P(Z<1.5)= = The probability that the chosen thermometer has a reading between 2.00 and 1.50 degrees is If many thermometers are selected and tested at the freezing point of water, then 91.04% of them will read between 2.00 and 1.50 degrees. 4-6 Normal Distribution 4-6 Inverse problem: Example 4-14, cont d Example 4-14 Let X=the current in milliamperes. X~N(10, 4) Step 1: Use Table III, find z=2.05 Step 2: Solve (x-10)/2=2.05 or equivalently x=10+2(2.05) = 14.1 milliamperes. Skip section

5 4-7-1 Normal Approximation to the Binomial and Poisson Distributions (skip) Figure 4-19 Normal approximation to the binomial Normal Approximation to the Binomial Distribution Example Normal Approximation to the Binomial Distribution Recall Z [ X E( X )]/ SD( X ) Normal Approximation to the Binomial Distribution Example P(X 151) Normal Approximation to the Binomial and Hypergeometric Distributions Normal Approximation to the Poisson Distribution 5

6 4-7-2 Normal Approximation to the Poisson Distribution Example Exponential Distribution Definition 950 or fewer P( X 950) P( X 950.5) P( Z ) P( Z 1.57) X 4-8 Mean and Variance 4-8 Example Lack of memory property 4-8 Exponential Distribution Example 4-22 (continued) Our starting point for observing the system does not matter. An even more interesting property of an exponential random variable is the lack of memory property. In Example 4-21, suppose that there are no log-ons from 12:00 to 12:15; the probability that there are no log-ons from 12:15 to 12:21 is still Because we have already been waiting for 15 minutes, we feel that we are due. That is, the probability of a log-on in the next 6 minutes should be greater than However, for an exponential distribution this is not true. 6

7 4-8 Exponential Distribution Example 4-22 (continued) 4-8 Exponential Distribution (skip) Example 4-21 (continued) 4-8 Exponential Distribution (skip) Example 4-21 (continued) 4-8 Mean and variance Example 4-21 (continued) nter Recap for Midterm 1 Midterm 1 covers Chapters 1-4 Probability rules and Bayes Theorem Cumulative Probability Function and percentile computation Discrete distributions: Probability mass function, Binomial, Geometric and Negative Binomial Continuous distributions: probability density function, uniform, normal and exponential Z-score Relationship between general normal and standard normal 4-9 Erlang Distribution Erlang Distribution The random variable X that equals the interval length until r counts occur in a Poisson process with mean λ > 0 has and Erlang random variable with parameters λ and r. The probability density function of X is for x > 0 and r =1, 2, 3,. 7

8 4-9 Gamma Function 4-9 Gamma Distribution 4-9 Erlang and Gamma Distributions 4-9 Gamma Distributions Gamma Distribution Gamma Distribution Figure 4-25 Gamma probability density functions for selected values of r and. =scale parameter If r=1, reduce to the exponential 4-10 Weibull Distribution Geometric Exponential X Negative Binomial Gamma(r, ) r Chi-squared if r= /2 Definition: Life time distribution Erlang if r=integer=>exponential if r=1 8

9 4-10 Weibull Distribution 4-10 Weibull Distribution Figure 4-26 Weibull probability density functions for selected values of and. It is named after Waloddi Weibull who described it in detail in First applied by Rosin & Rammler (1933) to describe the size distribution of particles 4-10 Weibull Distribution: Example Example Lognormal Distribution (skip) 4-11 Lognormal Distribution 4-11 Lognormal Distribution Example 4-26 Figure 4-27 Lognormal probability density functions with = 0 for selected values of 2. 9

10 4-11 Lognormal Distribution 4-11 Lognormal Distribution Example 4-26 (continued) Example 4-26 (continued) Geometric Exponential Negative Binomial Gamma(r, ) Chi-squared if r= /2 Erlang if r=integer=>exponential if r=1 10