Chemical Engineering Hand Written_Notes_Part_96

Chemical Engineering Hand Written_Notes_Part_96 - 194 5....

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194 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES In the univariate case, this reduces to familiar form (6.29) f ( ϕ )= 1 σ p (2 π ) exp ( ϕ ϕ ) σ 2 ¸ 6.2.2. Chi-square ( χ 2 ) distribution. This distribution is used in connection with Testing goodness of f t of experimental observations to hypothesized probability distributions Obtaining con f dence limits for the variance and the standard devia- tions Testing independence of variables. Let ( ϕ 1 , ..., ϕ m ) represent a set of m independent normally distributed ran- dom variables with parameters ( µ 1 2 1 ) , .... , ( µ m 2 m ) . If we calculate the squares of the standard normal variables (6.30) u 2 i = μ ϕ i ϕ i σ i 2 and sum the u 2 i s, then we have a new random variable χ 2 as follows (6.31) χ 2 = m X i =1 u 2 i Here, m is called degrees of freedom for χ 2 . The distribution of χ 2 depends only on m because u i are standardized. The probability density function for χ 2 can
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_96 - 194 5....

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