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Chapter 6

# Chapter 6 - IENG 213 Probability and Statistics for...

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IENG 213: Probability and Statistics for Engineers Instructor: Steven E. Guffey, PhD, CIH © 2002-2007

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2 Introduction Will cover most common continuous probability distributions Simplest distribution: Equal probability distribution (“continuous uniform distribution”) I.e., all events equally likely If there are values of equal probability in the interval [A,B], then: elsewhere B x A where A B B A x f 0 , 1 ) , ; ( = - = Also called “rectangular” distribution since the base and height are the same (1/[A-B])
3 Graph of Uniform Distribution The density function for a random variable on the interval [1,3] elsewhere B x A where A B B A x f 0 , 1 ) , ; ( = - = f(x) 0.5 1 3

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Example 6.1 Room can be reserved for < 4 hrs in any amount of time. Assume a uniformly distributed random variable, x. What is the prob density function?
5 Theorem 6.1 The mean and variance of the uniform distribution are: 12 ) ( 2 2 A B - = σ 2 B A + = μ

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6 6.2 Normal Distribution Most important continuous probability distribution Graph called the “normal curve” (bell-shaped) Derived by DeMoivre and Gauss. Called the “Gaussian” distribution. Describes many phenomena in nature, industry and research Random variable, X f(x) 2 3 5 4 6
Normal Distribution Definition: Density function of the normal random variable, X, with mean and variance such that: < < - = - - x e x n x , 2 ) , ; ( 2 ) ( 5 . 0 π σ σ μ σ μ 0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 -4.0 -2.0 0.0 2.0 4.0 Z-value f(x)

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8 Different Normal Distributions Different variances f(x) 0 1 3 2 4 Different means f(x) 0 1 3 2 4 5 6 Different means and variances f(x) 0 1 3 2 4
9 Normal Distributions f(x) Total area under the curve = 1 Symmetrical Inflection point at one std dev. Mean and mode Approaches 0 asymptotically

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10 6.3 Areas Under the Curve Area under the curve between x= x 1 and x= x 2 equals P(x 1 < x < x 2 ) dx e dx x n x X x P x x x x x - - = = < < 2 1 2 2 1 ) ( 5 . 0 2 1 2 1 ) , ; ( ) ( σ μ π σ σ μ f(x) x 1 x 2 2 P(x 1 < x < x 2 )
11 Standard Normal Distribution To enable use of tables, transform to unit values Mean = 0 Variance =1 σ μ - = X Z { } dz z n dz e dx e x X x P z z z z z x x x = = = < < - - - 2 1 2 1 2 2 1 2 ) 1 , 0 ; ( 2 1 2 1 ) ( 5 . 0 ) ( 5 . 0 2 1 π σ π σ σ μ μ z = 0 σ z = 1

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12 Definition 6.1 The distribution of a normal random variable with mean zero and variance = 1 is called a standard normal distribution. f(x) 2 µ σ x f(x) 2 0 σ z =1 P(x 1 < x < x 2 ) = P(z 1 < Z < z 2 ) P(x 1 < x < x 2 ) P(z1 < Z < z2)
13 Areas Under the Normal Curve z 0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 0.0600 0.0700 0.0800 0.0900 0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 Probability Z < z Z 2 P(Z < Z 2 )

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14 Example 6.2 Given the standard normal distribution, find the area under the curve that lies to the right of z = 1.84 Solution: (use Table A.3) f(x) Z=1.84 0 z 0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 P(z>1.84) = 1 - P(z < 1.84) = 1 – 0.9671 = 0.0329
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Chapter 6 - IENG 213 Probability and Statistics for...

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