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14Qualification

# 14Qualification - MAE 591 RANDOM DATA C W Lee Probability...

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MAE 591 RANDOM DATA C. W. Lee Probability Distribution Function 1. Normal distribution for standardized variable x x x z σ µ = with 0 = µ z and σ . 1 2 = z 2 2 2 1 ) ( z e z p π = , π = z u du e z P 2 2 2 1 ) ( 100 α percentage point : α α α α α π = > = α α = = = z z z dz e z z z z dz z p z P 2 2 2 1 ] Pr[ 1 ] Pr[ ) ( ) ( ) ( z p 0.4 α z α 1 0.1587 2 0.0228 3 0.0013 α z α 4 3 2 1 0 -1 -2 -3 -4 0.3 0.2 0.1 z 0 Three basic distributions associated with the normal distribution are chi-square, t- and F- distributions . These distributions are frequently used to answer as: whether an assumed type of probability distribution fits the observational facts and whether two or more variables are similarly distributed or more specifically, whether two or more variables have a common average value. 1

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MAE 591 RANDOM DATA C. W. Lee 2. Chi-square distribution A new random variable is defined as 2 2 2 2 1 2 n n z z z + + + = χ " where are n independent normal distributed random variables with zero mean and unit variance. χ is called the chi-square variable with n DOF. Then the probability density function of is given by n z z z z , , , , 3 2 1 " 2 n 2 n χ () 2 1 2 2 1 2 2 2 2 2 ) ( χ χ Γ = χ e n p n n where Γ , 0 , is Gamma function. = dx e x z x z 1 ) ( Re > z 100 percentage point : α α = χ > χ = χ χ χ α α 2 ; ] Pr[ ) ( 2 ; 2 2 2 n n d p 0 2 n χ ) ( 2 n p χ n = 8 n = 4 n = 2 n = 1 0.5 0.4 0.3 0.2 0.1 0 20 18 16 14 12 10 8 6 4 2 Note: As , n 2 2 n χ becomes normal distributed with mean of 1 2 n
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14Qualification - MAE 591 RANDOM DATA C W Lee Probability...

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