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Lecture7_Basic Probability

Lecture7_Basic Probability - Lecture 8 Basic Probability...

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Lecture 8 Basic Probability Concepts
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Normal Distribution ( 29 2 2 2 1 P( ) 2 where is the mean value of the variable and is the standard deviation of the variable Probability density function for a normal distribution is give . y n b y y y y y y y e μ σ σ π μ σ - - = P( ) y y y σ - y σ μ
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Web Based Probability Calculator
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Transformation From The Normal Distribution To The Standard Normal Distribution Assume we have a of the variable with: mean value of standard de Normal Distribution Standard Normal Distr viation of The is the transformation from to via: ibution Th e y y y y y y y y z y z μ σ μ σ = = - = Standard Normal Distribution has a mean of zero and a standard deviation of 1!
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* * What is the probability , where is some specific value of ? y y y y * * Pr( ) Pr( ) y y z z =
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Z Table (Posted on WebCT)
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Z Table Example - z* = 0.75
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Some Z Table Values (Table 5.1 of Hyman) z* One sided probability Two sided probability 0 0.5 1.0 0.5 0.308 0.616 1.0 0.159 0.318 1.5 0.067 0.134 2.0 0.023 0.046 2.5 0.0062 0.0124 3.0 0.00135 0.0027 1 σ 2 σ 3 σ
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USL, LSL, and Tolerance LSL = Lower specification limit, USL = Upper specification limit Standard tolerance values assume: Tolerance 3 or 3 y y y y σ σ = ± ∆ = ± = μ LSL -3 σ USL 3 σ
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Resistor Example ( 29 4 A batch of resistors is specified as 10 Kohm 5% Find the tolerance R, and . R 10 0.05 500 ohms 10 Kohm Assume the tolerance, R, represents 3 , which implies that for a large batch of resistors, al μ σ μ σ ± = = = l but 0.3% will fall between 9.5 Kohm and 10.5 Kohm, and R 500 167 ohms 3 3 σ = = =
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Central Limit Theorem 1 2 3 i i 1 2 3 Let , , ... be a set if random variables for which both the expected (mean) values, , and standard deviations, , exist and are finite. Assume is a function of , , ... . The c n n x x x x f x x x x μ σ entral limit theorem states that the probability distribution of will converge to a normal distribution as becomes large. f n
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History of Central Limit Theorem The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born English mathematician Abraham de Moivre, who, in a remarkable article published in 1733 , used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician Pierre-Simon Laplace rescued it from obscurity in his monumental work Théorie Analytique des Probabilités, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901 , Russian mathematician Aleksandr Lyapunov defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit
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