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LectureNotes8 - Continuous Random Variable and Their...

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Continuous Random Variable and Their Probability Distributions: Part II Cyr Emile M’LAN, Ph.D. [email protected] Continuous Data Models: Part II – p. 1/30

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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 , 0 , elsewhere . We write X ∼ U [ a, b ] . The cumulative distribution function of X is F ( x ) = integraldisplay x -∞ f ( u ) du = 0 , x < a , x - a b - a , a x b , 1 , x > b . Continuous Data Models: Part II – p. 3/30

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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 , 0 x < 360 0 , 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 0 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 0 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

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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 0 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 ) . Continuous Data Models: Part II – p. 6/30
Joke Old statisticians never die, they just undergo a transformation. Continuous Data Models: Part II – p. 7/30

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The Exponential Distribution Density Function A random variable X is said to have an exponential distribution with parameter θ > 0 if it has density f ( x ) = 1 θ e - x/θ , if x > 0 , 0 , otherwise .
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LectureNotes8 - Continuous Random Variable and Their...

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