Chapter 10 Continuous probability distributions - Ugrad.math.ubc.ca

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$Math 103, Section 203, Spring Term 2004, Midterm # 1 (35 min ...$

Math 103 Notes Chapter 10up to the second are usefully employed to describe common attributes of a distribution.10.6.1 Moments of a probability density distributionIn the particular case that the distribution is a probability density, p(x), defined on the intervala ≤ x ≤ b, we have already established the following :M 0 =∫ bap(x) dx = 1(This follows from the basic property of a probability density.) ThusThe zero’th moment of any probability density is 1.FurtherM 1 =∫ bax p(x) dx = ¯x = µ.That is,The first moment of a probability density is the same as the mean (i.e. expected value) of thatprobability density.So far, we have used the symbol ¯x to represent the mean or average value of x but often thesymbol µ is also used to denote the mean.The second moment, of a probability density also has a useful interpretation. From abovedefinitions, the second moment of p(x) over the interval a ≤ x ≤ b isM 2 =∫ bax 2 p(x) dxWe will shortly see that the second moment helps describe the way that density is distributed aboutthe mean. For this purpose, we must describe the notion of variance or standard deviation.Variance and standard deviationTwo kids of approximately the same size can balance on a teeter-totter by sitting very close to thepoint at which the beam pivots. They can also achieve a balance by sitting at the very ends of thebeam, equally far away. In both cases, the center of mass of the distribution is at the same place:precisely at the pivot point. However, the mass is distributed very differently in these two cases.In the first case, the mass is clustered close to the center, whereas in the second, it is distributedfurther away. We may want to be able to describe this distinction, and we could do so by consideringhigher moments of the mass distribution.Similarly, if we want to describe how a probability density distribution is distributed about itsmean, we consider moments higher than the first. We use the idea of the variance to describewhether the distribution is clustered close to its mean, or spread out over a great distance from themean.v.2005.1 - January 5, 2009 16
Math 103 Notes Chapter 10VarianceThe variance is defined as the average value of the quantity (distance from mean) 2 , where theaverage is taken over the whole distribution. (The reason for the square is that we would not likevalues to the left and right of the mean to cancel out.)For discrete probability with mean, µ we define variance byV = ∑ (x i − µ) 2 p iFor a continuous probability density, with mean µ, we define the variance byV =∫ ba(x − µ) 2 p(x) dxThe standard deviationThe standard deviation is defined asσ = √ VLet us see what this implies about the connection between the variance and the moments of thedistribution.Relationship of variance to second moment¿From the equation for variance we calculate thatV =∫ baExpanding the integral leads to:(x − µ) 2 p(x) dx =∫ ba(x 2 − 2µx + µ 2 ) p(x) dx.V ==∫ ba∫ bax 2 p(x)dx −∫ bx 2 p(x)dx − 2µa∫ b2µx p(x) dx +a∫ bx p(x) dx + µ 2 ∫ baµ 2 p(x) dxap(x) dx.We recognize the integrals in the above expression, since they are simply moments of the probabilitydistribution. Plugging in these facts, we arrive atv.2005.1 - January 5, 2009 17
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Chapter 10 Continuous probability distributions - Ugrad.math.ubc.ca

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