ECE 340 Probabilistic Methods in Engineering M/W 3-4:15. Lecture 10: Continuous RV Families. Prof. Vince Calhoun

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1 ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Lecture 10: Continuous RV Families Prof. Vince Calhoun 1

2 Reading This class: Section Next class: Section

3 Homework 3.9, 3.49, 4.5, 4.12, 4.14 Plus problems on handout 3

4 Section Outline Important continuous RV s Uniform Exponential Gaussian Gamma Rayleigh Cauchy laplacian Functions of RV s 4

5 Common families of Continuous Distribution As mentioned previously, the development of the probability distribution of a continuous random variable is not as intuitive as in the discrete case. Thus, we will concentrate on exploring several common families of distributions, which fit a wide range of experiments 5

6 The Uniform Distribution o The simplest distribution is the one in which a continuous r.v. can assume any value within a interval [a, b] Def: A continuous r.v. X is said to have a uniform distribution on the interval [a,b] if the probability distribution (pdf) of X is: 1 a x b f ( x) = b a 0 otherwise 6

7 7 The Uniform Distribution The cumulative distribution is 12 ) ( ) ( 2 ) 1 ( ) ( ] [ ) ( ) ( ) ( ) ( 2 a b X V a b dx a b x dx x xf X E a b a x a b a a b x a x a b x dx x f dx x f x X P X F x x x x = + = = = = = = = = =

8 The Uniform Distribution Note: An important uniform distribution is that for when a = 0 and b = 1, namely U(0, 1) A U(0,1) r.v. can be used to simulate observation of other random variables of the discrete and continuous type. 8

9 Exponential Distribution Previously, we discussed the Poisson random variable, which was the number of events occurring in a given interval. This number was a discrete r.v. and the probabilities associated with it could be described by the Poisson Probability Distribution. Not only is the number of events a r.v., but the waiting time between event is also a random variable. This r.v. is a continuous r.v. for it can assume any positive value. This r.v. is an exponential r.v. which can be described by the exponential distribution. 9

10 10 Exponential Distribution i. Pdf: > = otherwise x e x f x & ) ( λ λ λ where λ = rate at which events occur ii. Correspondingly, ) ( 1 ] [ 0, 1 ) ( ) ( λ λ λ λ λ = = = = = X V X E x e dx e x X P x F x x x iii. An important application of the exponential distribution is to model the distribution of component lifetime. A reason for its popularity is because of the memory-less property of the Exponential Distribution

11 Exponential Distribution An example: A production line has the potential to break down, with an average time between breakdown events of 10 months (e.g. λ = 0.10 /month). What is the probability of the time between breakdowns being one year or less? P ( breakdown year ) = e 0.1 x dx P ( breakdown year ) = 1 e 0.1(12 ) =

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29 Normal Distribution It is a fact that measurements on many random variables will follow a bellshaped distribution. Random variable of this type are closely approximated by a Normal Probability Distribution. A continuous r.v. X is said to have a normal distribution if the pdf of X is f ( x) = 1 e 2πσ ( x μ ) 2 2σ 2, σ > 0, < x <, < μ < The distribution contains 2 parameters (μ and σ). These are the expected value and the variance and hence locate the center of the distribution and measure its spread. 29

30 Normal Distribution The Standard Normal Distribution To compute P(a x b) when X ~ N(μ, σ 2 ), we must evaluate b a f b ( x μ ) 2 2σ ( x) dx = a 1 e 2πσ 2 dx Note: None of the standard integration techniques can be used to evaluate this pdf. Instead, for μ = 0, and σ 2 = 1, the pdf has been evaluated and values have been computed. Using the table, probabilities for any other values of μ and σ 2 can be determined 30

31 Normal Distribution The normal distribution for parameters values μ = 0, and σ 2 = 1 is called the standard normal distribution. A r.v. that has a standard distribution is called a standard normal random variable (denoted by Z). The pdf of Z is: f z 1 2 ( z) = e 2π 2, < z < 31

32 Normal Distribution The cumulative distribution of Z is z P ( Z z) = f ( y) dy and is denoted by Φ(Z) Note: The N(0,1) Table returns the cumulative probability up to z or Φ(z) 32

33 Normal Distribution Non-standard Normal Distribution The table only provides probabilities for r.v. following the N(0,1) distribution. Thus, when X ~ N(μ, σ 2 ), (i.e. not μ = 0, σ 2 = 1), probabilities involving X are computed by standardizing the r.v. to N(0,1) scale. 33

34 Normal Distribution The Standardized Variable is x μ σ Subtracting the μ shifts the mean from μ to zero, and then dividing by σ scales the variable, so that the standard deviation is one rather than σ. 34

35 35 Normal Distribution Def: If X has a normal distribution with mean μ and variance σ 2 [X~N(μ, σ 2 )] then: σ = x μ Z is a standard normal r.v. [Z~N(0,1)] Note: ) ( ) ( ) ( ) ( σ μ σ μ σ μ σ μ = Φ = = x x Z P x X P x X P

36 Normal Distribution An example of converting to standard normal distribution is given by the data from a dry plasma etch study (Lynch and Markle, 1997). The data are in angstroms, from the before process improvement trial. The mean is , and the standard deviation is angstroms. Run Value Z- value

37 Normal Distribution This example demonstrated the yielded values from the normalization equation for Z. The values obtained can then be used to compare the likelihood of occurrence when comparing to other data. This is done in hypothesis testing and related methods (SPC, etc.) 37

38 Normal approximates to Binomial Let X be a binomial random variable based on n trials with probability of success p. If the histogram of the binomial probability is not too skewed, X has approximate a normal distribution with μ = np and σ 2 = npq In particular X ~ Bin(n, p) X np P( X x) = Φ( ) npq In practice, the approximate is adequate provided np 5and nq 5 38

39 Multivariate/ Joint Probability Distributions Up to now, we have studied probability models for a single random variable (either discrete or continuous). Many applied problems involve the combination of several random variables. Let X and Y be 2 random variables. The joint probability mass function p(x, y) is defined for each pair of number (x, y) by P(x, y) = P(X = x, and Y = y) Joint probability is merely on extension of the techniques that you learned in the single random variable case. 39

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