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 10ThusV = M 2 − 2µ µ + µ 2V = M 2 − µ 2Thus the variance is clearly related to the second moment and to the mean of the distribution.Relationship of variance to second moment¿From the definitions given above, we find thatσ = √ V = √ M 2 − µ 210.6.2 ExampleConsider the continuous distribution, in which the probability is constant, p(x) = C, for values of xin the interval [a, b] and zero for values outside this interval. Such a distribution is called a uniformdistribution. (It has the shape of a rectangular band of height C and base (b − a).) It is easy tosee that the value of the constant C should be 1/(b − a) so that the area under this rectangularband will be 1, in keeping with the property of a probability distribution. Thus the equation of thisprobability is p(x) = 1 . We compute the first moments of this probability densityb − aM 0 =∫ bap(x)dx = 1b − a∫ ba1 dx = 1.(This was already known, since we have determined that the zeroth moment of any probabilitydensity is 1.) We also find thatM 1 =∫ bx p(x) dx = 1 x dxb − a a= 1x 2bb − a 2 ∣ = b2 − a 22(b − a) .This last expression can be simplified by factoring, leading toaµ = M 1 =a(b − a)(b + a)2(b − a)∫ b= b + a2Thus we have found that the mean µ is in the center of the interval [a, b], as expected. The medianwould be at the same place by a simple symmetry argument: half the area is to the left and halfthe area is to the right of this point.v.2005.1 - January 5, 2009 18
Math 103 Notes Chapter 10To find the variance we might first calculate the second moment,M 2 = x 2 p(x) dx = 1 ∫ bx 2 dxab − a aIt can be shown by simple integration that this yields the result∫ bwhich can be simplified toM 2 = b3 − a 33(b − a) ,M 2 = (b − a)(b2 + ab + a 2 )3(b − a)= b2 + ab + a 2.3We would then compute the varianceAfter simplification, we getThe standard deviation is thenV = M 2 − µ 2 = b2 + ab + a 2−3V = b2 − 2ab + a 212σ ==(b − a)2 √ 3 .(b − a)2.12(b + a)2.4v.2005.1 - January 5, 2009 19
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Chapter 10 Continuous probability distributions - Ugrad.math.ubc.ca

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