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Week 15 Wed Dec 8

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

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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 .

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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
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.

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