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Unformatted text preview: Using Transformations Generating from a Normal distribution (BoxM¨uller Method): We need two sequences of standard uniform variables. Let U 1 and U 2 be two independent standard uniform variables. Define Z 1 := [ 2 ln U 1 ] 1 / 2 cos (2 πU 2 ) Z 2 := [ 2 ln U 1 ] 1 / 2 sin (2 πU 2 ) Then both Z 1 and Z 2 have a standard normal distribution and are independent. To get X i ∼ N ( μ, σ 2 ) transform using X i := μ + σZ i , i = 1 , 2 To get a sequence of n random numbers from a Normal N ( μ, σ 2 ) distribution, we use a U (0 , 1) sequence u i , i = 1 , . . . , n , convert them to z i , i = 1 , . . . , n and use them to compute x i , i = 1 , . . . , n . 1 Inverse CDF Method for Continuous Densities Main Idea: Consider generating a random variable X from a distribution with cdf F and suppose that F is continuous and strictly increasing and F 1 ( u ) is welldefined for ≤ u ≤ 1 . If U is a random variable from U (0 , 1) , then it can be shown that X = F 1 ( U ) has distribution F . x F X x o 2 Applications of Inverse CDF Method This result can be used to obtain closed form transformations only if F can be inverted analytically i.e., the equation u = F ( x ) can be solved for x in closed form. However, if F can be accurately and efficiently inverted numerically, this method can be very useful....
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 Spring '11
 Zhou
 Normal Distribution, Probability theory, Exponential distribution, probability density function, Cumulative distribution function, λ

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