Lecture#04-06.pdf - Contents 14 Uniform Distribution 1 14.1...

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Contents 14 Uniform Distribution 1 14.1 Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 14.2 Expectation & Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 14.3 Uniform in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 14.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 15 Normal Distribution 3 15.1 Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 15.2 Expectation & Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 15.3 Normal in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 15.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 14 Uniform Distribution 14.1 Probability Density Function Definition : uniform distribution For θ 1 < θ 2 , the continuous random variable Y Uniform ( θ 1 , θ 2 ) is said to have a uniform distribution on interval ( θ 1 , θ 2 ) if and only if its pdf is f ( y ) = 1 θ 2 - θ 1 , θ 1 y θ 2 0 , otherwise For example, the following is a plot of the pdf for Uniform (3 . 2 , 4 . 7) . 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 x f(x) θ 1 θ 2 Notes.
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The uniform distribution has 2 parameters, θ 1 and θ 2 that define the lower and upper limit of the range of Y . All allowable values of the random variable, i.e. all points in ( θ 1 , θ 2 ) , are equally probable. The uniform distribution is important for two reasons: – Random number generation. If a computer program can generate U Uniform (0 , 1) (e.g. runif(1) ), then Y = F - 1 ( U ) F ( y ) . In other words, once a computer can generate a uniform random variables, it can also generate random variables Y from any distribution F ( y ) if the inverse function F - 1 ( · ) is available. – Many physical phenomena have approximate uniform distribution. If events occur as a Pois- son process, then given an event has occurred in the interval ( a, b ) , the exact time or location of that event has a Uniform ( a, b ) distribution. F ( y ) = R y a 1 b - a dt = t b - a y a = y - a b - a . The standard uniform random variable is Y Uniform (0 , 1) , with f ( y ) = 1 . Relation to Poisson To be more concrete about the relationship to the Poisson, we will prove the following theorem about the time/location of an event after it is known that an event has occurred. (Recall, that the Poisson random variable arises in physical situations where events are happening randomly within some inteval in time or space.) Theorem 38. Suppose Y Poisson ( λ ) on interval ( a, b )
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