Lecture3 - PROBABILITY DISTRIBUTIONS: DISCRETE AND...

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PROBABILITY DISTRIBUTIONS: DISCRETE AND CONTINUOUS Univariate Probability Distributions. Let S be a sample space with a prob- ability measure P defned over it, and let x be a real scalar-valued set Function defned over S , which is described as a random variable. IF x assumes only a fnite number oF values in the interval [ a, b ], then it is said to be discrete in that interval. IF x assumes an infnity oF values between any two points in the interval, then it is said to be everywhere continuous in that interval. Typically, we assume that x is discrete or continuous over its entire domain, and we describe it as discrete or continuous without qualifcation. IF x ∈{ x 1 ,x 2 ,... } , is discrete, then a Function f ( x i ) giving the probability that x = x i is called a probability mass function. Such a Function must have the properties that f ( x i ) 0 , For all i, and X i f ( x i )=1 . Example. Consider x 0 , 1 , 2 , 3 } with f ( x )=(1 / 2) x +1 . It is certainly true that f ( x i ) 0 For all i . Also, X i f ( x i )= ½ 1 2 + 1 4 + 1 8 + 1 16 + ··· ¾ =1 . To see this, we may recall that 1 1 θ = { 1+ θ + θ 2 + θ 3 + ···} , whence θ 1 θ = { θ + θ 2 + θ 3 ++ θ 4 + . Setting θ / 2 in the expression above gives θ/ (1 θ 1 2 / (1 1 2 , which is the result that we are seeking. IF x is continuous, then a probability density function (p.d.F.) f ( x ) may be defned such that the probability oF the event a<x b is given by P ( b Z b a f ( x ) dx. Notice that, when b = a , there is P ( x = a Z a a f ( x ) dx =0 . That is to say, the integral oF the continuous Function f ( x ) at a point is zero. This corresponds to the notion that the probability that the continuous random variable 1
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x assumes a particular value amongst the non denumerable (i.e. uncountable) in- fnity oF values within its domain must be zero. Nevertheless, the value oF f ( x )at a point continues to be oF interest; and we describe this as a probability measure as opposed to a probability. Notice also that it makes no di±erence whether we include oF exclude the endpoints oF the interval [ a, b ]= { a x b } . The reason that we have chosen to use the halF-open halF-closed interval ( a, b { a<x b } is that this accords best with the defnition oF a defnite integral over [ a, b ], which is obtained by subtracting the integral over ( −∞ ,a ] From the integral over ( −∞ ,b ]. Example. Consider the exponential Function f ( x )= 1 a e x/a defned over the interval [0 , ) and with α> 0. There is f ( x ) > 0 and Z 0 1 a e x/a dx = h e x/a i 0 =1 . ThereFore, this Function constitutes a valid p.d.F. The exponential distribution provides a model For the liFespan oF an electronic component, such as Fuse, For which the probability oF Failing in the ensuing period is liable to be independent oF how long it has survived already. Corresponding to any p.d.F f ( x ), there is a cumulative distribution Function, denoted by F ( x ), which, For any value x , gives the probability oF the event x x .
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This note was uploaded on 03/02/2012 for the course EC 2019 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.

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Lecture3 - PROBABILITY DISTRIBUTIONS: DISCRETE AND...

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