10-More Continuous Distributions

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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern