notes_03 - probability distributions Uniform distribution...

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MIT OpenCourseWare http://ocw.mit.edu 1.010 Uncertainty in Engineering Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu.terms .
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± ± ± 1.010 - Brief Notes # 3 Random Variables: Continuous Distributions Continuous Distributions Cumulative distribution function (CDF) F X ( x ) = P [ X x ] P [ x 1 < X x 2 ] = F X ( x 2 ) F X ( x 1 ) Average probability density in an interval [ x 1 ,x 2 ] P [ x 1 < X x 2 ] F X ( x 2 ) F X ( x 1 ) = x 2 x 1 x 2 x 1 Probability density function (PDF) f X ( x 1 ) = lim P [ x 1 < X x 2 ] = dF X x 2 x 1 x 2 x 1 dX x 1 x 2 f X ( x ) dx = P [ x 1 < X x 2 ] = F X ( x 2 ) F X ( x 1 ) x 1 Properties of the PDF 1. f X ( x ) 0 2. f X ( x ) dx = 1 −∞ u 3. f X ( x ) dx = F X ( u ) −∞ 1
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2 Example of PDF and corresponding CDF of a continuous random variable: steel-yield-stress. (a) Probability density function; (b) cumulative distribution function. Examples of continuous
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Unformatted text preview: probability distributions Uniform distribution c, where c is constant, a &lt; x &lt; b f X ( x ) = , otherwise. 1 Then c = b a , x &lt; a x a , b a a x b 1 , x &gt; b F X ( x ) = Example of uniform distribution. 3 Exponential Distribution Let T = time to frst arrival in a Poisson point process F T ( t ) = P [ T t ] = 1 P [ T &gt; t ] = 1 P [no occurrence in [0, t]] = 1 e t f T ( t ) = e t , t PDF and CDF o the exponential distribution....
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