{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Week 15 Wed Dec 8 - Posted before class WEEK 15 Wednesday...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Posted before class. WEEK 15, Wednesday, Dec 8 Section 3.10 . Continuous Random Variables . Discuss and contrast discrete random variables with their probability mass functions with continuous random variables with their probability density functions. Continuous Probability density function f ( x ) Cumulative distribution function x dy y f x X P x F ) ( ) ( ) ( ) ( [ ) ( ] ) [( ) ( ) ( ) ( 2 2 2 2 2 2 dx x f x X E X E X V dx x xf X E ) ( ) ( X V X SD Mathematical models for so- called “ continuous random variables ” are very different from those for discrete random variables. Continuous random variables are modeled to take values on the real number line in this way. The probability distribution is defined through the probability density function for the random variable (see page 128 of the textbook). Consider a random variable X with probability density function f . Then here are the properties of f . f ( x ) ≥ 0 for all - ∞ < x < ∞ 1 ) ( dx x f and b a dx x f b X a P ) ( ) ( for all real numbers a < b . With this model for probability distributions, probabilities are given by areas and ) ( ) ( ) ( ) ( ) ( b X a P b X a P b X a P dx x f b X a P b a .
Image of page 1

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

View Full Document Right Arrow Icon
2 It follows that 0 ) ( ) ( ) ( a a dx x f a X a P a X P for all real numbers a . Individual points get zero probability when using probability density functions! For any p with 0 < p < 1, one can find a 100pth percentile of the distribution, that is, a point x 0 such that P(X < x 0 ) = p and P(X > x 0 ) = 1 p. Exercise 3-79 . (c) Ans. Section 3.11 . Continuous Uniform on the Interval ( A , B ) . Here is the density function: A B 1 , A < x < B ) ( x f 0, otherwise Here is a graph of the density function. y = f ( x ) A B x Important Results for U ( A , B ): = E ( X ) = ( A + B )/2 2 = V ( X ) = ( B A ) 2 /12 , = SD ( X ) = ( B A )/ 12. Exercise 3-78 . Ans. 3/4 = 0.75; = 7 minutes
Image of page 2
3 Section 3.12 . Exponential Distribution with Parameter > 0 . Here is the density function: x e , x ≥ 0 ) ( x f 0, x < 0 Important Results for Exponential ( ): = E ( X ) = 1/ 2 = V ( X ) = 1/ 2 , = SD ( X ) = 1/ . There are some useful formulas for probabilities of events involving an exponentially distributed random variable X . Consider these useful formulas (p. 131): x x t x t e e dt e x X P x F 1 | ) ( ) ( 0 0 , x > 0 x e x X P ) ( , x > 0 b a e e b X a P ) ( , 0 ≤ a b Example 1. Light Bulbs . A plant has a large assembly area with many light bulbs that are turned on 24 hours per day. An electronic device records how many hours each new light bulb burns until it fails and finds across a large sample that the mean time to failure for new light bulbs is 2,732 hours. Based on the failure time data, the plant engineer models the failure time X of a new light bulb as Exponential( ) with mean 2,732 hours. Use this model to answer the following questions.
Image of page 3

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

View Full Document Right Arrow Icon
Image of page 4
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