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Unformatted text preview: 1 Continuous Random Variables In this lecture we introduce a number of continuous random variables that are useful in practice. 1.1 The Uniform Distribution We say that U has a uniform distribution over the interval [0 , 1] if each subinterval of the same length has the same probability of being selected. Thus f ( x ) must be a constant over [0 , 1] and zero outside [0 , 1] and the constant must be equal to one for the integral to be one. Thus the density of the U ∼ [0 , 1] random variable is f ( x ) = 1 0 ≤ x ≤ 1 , and f (0) = 0 elsewhere, and the cdf is F ( x ) = x over 0 ≤ x ≤ 1. It is easy to see that E [ U ] = 1 / 2 and that Var( U ) = 1 / 12. Suppose U is uniform [0 , 1]. Then V = a + ( b- a ) U is uniform over the interval [ a,b ]. It follows that EV = a + ( b- a ) / 2 = ( a + b ) / 2 and that Var( V ) = ( b- a ) 2 / 12. Example: Suppose the arrival time of a buss at a bus stop is uniformly between 7 : 15 and 7 : 22 . What is the probability that it has already arrived by 7 : 20 . ? Remark: We can afford to be sloppy with the end points when dealing with continuous random variables. In particular, the he density will still be well defined if it is over the intervals ( a,b ] , [ a,b ), or ( a,b ). 1.2 The Normal Distribution The normal distribution is the most important distribution in probability and statistics because averages of random variables have a density that converge to a bell shaped curve that is symmetric around the mean and is known as the normal density. The normal distribution, also called the Gaussian distribution, was used in German bank notes before they were replaced with the Euro. We start our discussion with the standard normal density and then build up to general normal densities. We say that Z has the standard normal density with parameters 0 and 1 if φ ( z ) = 1 √ 2 π e- z 2 / 2- ∞ < z < ∞ . (1) Notice that φ ( z ) ≥ 0. It can be verified that the standard density integrates to one and that E [ Z ] = 0 and Var( Z ) = 1, thus the parameters of the normal coincide with the mean and variance....
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- Spring '04
- Normal Distribution, Probability theory, tk