10-More Continuous Distributions

# 10-More Continuous Distributions - IE 111 Fall Semester...

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IE 111 Fall Semester 2009 Note Set #10 Weibull, log-normal, hazard functions While the Normal distribution is the most commonly used continuous distribution, there are many other distributions that prove useful in engineering as well as other fields. Indeed I sometimes get the impression that students think there is only one continuous distribution, or that all continuous distributions look “bell shaped”. Nothing could be further from the truth. Thus we now study some additional useful distributions. You should note carefully the shape of the distribution, and the uses that can be made of it. The Log-Normal Distribution The Log-Normal distribution is related to the Normal distribution as follows: If X is a Normally distributed random variable, and Y = e X , then Y is a Log-Normal random variable. Similarly, X = Ln(Y) is Normal if Y is Log-Normal. The density function of the Log-Normal distribution is: f X x = 1 x 2 e ln x − 2 2 2 , for x>0 You should look at the graph of the Log-Normal density on page 148. It is a heavily skewed distribution, and as such is useful for modeling random variables with similar behavior. The Log-Normal; has 2 parameters, μ and σ . The mean is E(X) = EXP[ μ + σ 2 /2] The Variance is V(X) = EXP[2 μ + σ 2 ] ( EXP[ σ 2 ] -1 ) The CDF has no closed form solution, but we can use the Normal tables to compute probabilities associated with the Log-Normal distribution. Again let Y be a Log-Normal random variable and let X = ln(Y) so that X is Normally distributed. Thus P(a < Y < b) = P ( ln(a) < X < ln(b) ) = Φ ( ( ln(b)- μ ) / σ ) - Φ ( ( ln(a)- μ ) / σ ) Were Φ is the standard Normal CDF.

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