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LectureNotes8 - Continuous Random Variable and Their...

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Unformatted text preview: Continuous Random Variable and Their Probability Distributions: Part II Cyr Emile MLAN, Ph.D. mlan@stat.uconn.edu Continuous Data Models: Part II p. 1/30 Some Common Continuous Random Variables Text Reference : Introduction to Probability and Its Application, Chapter 5. Reading Assignment : Sections 5.3-5.5, 5.7, March 18 - March 22 In this set of note, we will discuss several common but useful distribution models for continuous random variables. Continuous Data Models: Part II p. 2/30 The Uniform Distribution A random variable X is uniformly distributed over the unit interval [ a,b ] (a<b) if its probability density function is given is by f ( x ) = 1 b- a a x b, , elsewhere . We write X U [ a,b ] . The cumulative distribution function of X is F ( x ) = integraldisplay x- f ( u ) du = , x < a, x- a b- a , a x b, 1 , x > b. Continuous Data Models: Part II p. 3/30 The Uniform Distribution Example 5.8 : The direction of an imperfection with respect to a reference line on a circular object such a tire, brake rotor, or flywheel is, in general, subject to uncertainty. Consider the reference line connecting the valve stem on a tire to the center point, and let X be the angle measured clockwise to the location of an imperfection. One possible probability density function for X is f ( x ) = 1 360 , x < 360 , elsewhere a). Find the probability that the angle of occurrence lies within 90 o and 180 o . b). Find the probability that the angle of occurrence lies within o and 90 o . Continuous Data Models: Part II p. 4/30 The Uniform Distribution c). Find the probability that the angle of occurrence is within 90 o of the reference line . Solution : a). P (90 X 180) = integraldisplay 180 90 1 360 dx = x 360 vextendsingle vextendsingle vextendsingle x =180 x =90 = 1 4 . b). P (0 X 90) = integraldisplay 90 1 360 dx = x 360 vextendsingle vextendsingle vextendsingle x =90 x =0 = 1 4 . c). P (0 X 90) + P (270 X < 360) = 1 4 + 1 4 = 1 2 . Continuous Data Models: Part II p. 5/30 The Uniform Distribution The mean and variance are respectively E [ X ] = a + b 2 and Var ( X ) = ( b- a ) 2 12 . The moment generating function of X is M X ( t ) = integraldisplay b a e tx b- a dx = e bt- e at ( b- a ) t if t negationslash = 0 and M X (0) = 1 . There is a relationship between the uniform distribution and the Poisson distribution. Suppose that the number of events that occur in an interval (0 ,t ) has a Poisson distribution. If exactly one of these events is known to have occurred in the interval ( a,b ) with a < b t , then the conditional probability distribution of the actual time of occurrence for this event given it has occurred is uniform over ( a,b ) ....
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LectureNotes8 - Continuous Random Variable and Their...

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