notes509fall11sec34

notes509fall11sec34 - STAT 509 Section 3.4: Continuous...

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Unformatted text preview: STAT 509 Section 3.4: Continuous Distributions Probability distributions are used a bit differently for continuous r.v.s than for discrete r.v.s. A continuous random variable is one for which the outcome can be any value in an interval of the real number line. Examples Let Y = length in mm Let Y = time in seconds Let Y = temperature in C Continuous distributions typically are represented by a probability density function (pdf), denoted f ( y ). Properties of Density Functions: (1) f ( y ) 0 for all possible y values. (Density function always on or above the horizontal axis) (2) - dy y f ) ( = 1. (Total area beneath the curve is exactly 1.) (3) The cumulative distribution function (cdf) is again denoted by F( y ). F( y ) = P(Y < y ) = - y dt t f ) ( (4) For continuous r.v.s, the probability distribution will give us the probability that a value falls in an interval (for example, between two numbers). That is, the probability distribution of a continuous r.v. Y will tell us P( y 1 Y y 2 ), where y 1 and y 2 are particular numbers of interest. Specifically, P( y 1 Y y 2 ) is the area under the density function between y = y 1 and y = y 2 : P( y 1 Y y 2 ) = 2 1 ) ( y y dy y f = F( y 2 ) F( y 1 ) Example Picture: Example 1:...
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notes509fall11sec34 - STAT 509 Section 3.4: Continuous...

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