It is equally valid for functions that are piecewise

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transformations. It is equally valid for functions that are piecewise monotonic, i.e. ones that have monotonic segments, which includes virtually all of the functions that we might consider. 7
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The moments of a distribution. The r th raw moment of x relative to the datum a is defined by the expectation E { ( x a ) r } = Z x ( x a ) r f ( x ) dx if x is continuous, and E { ( x a ) r } = X x ( x a ) r f ( x ) if x is discrete . The variance, which is a measure of the disperion of x is the second moment of about the mean. This measure is minimise by the choice of a = E ( x ): V ( x ) = E [ { x E ( x ) } 2 ] = E [ x 2 2 xE ( x ) + { E ( x ) } 2 ] = E ( x 2 ) { E ( x ) } 2 . We can also define the variance operator V . (a) If x is a random variable, then V ( x ) > 0. (b) If a is a constant, then then V ( a ) = 0. (c) If a is a constant and x is a random variable, then V ( ax ) = a 2 V ( x ). To confirm the latter, we may consider V ( ax ) = E { [ ax E ( ax )] 2 } = a 2 E { [ x E ( x )] 2 } = a 2 V ( x ) . If x, y are independently distributed random variables, then V ( x + y ) = V ( x ) + V ( y ). But this is not true in general. 8
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The variance of the binomial distribution. Consider a sequence of Bernoulli trials with x i { 0 , 1 } for all i . The p.d.f of the generic trial is f ( x i ) = p x i (1 p ) 1 x i . Then E ( x i ) = X x i x i f ( x i ) = 0 . (1 p ) + 1 .p = p. It follows that, in n trials, the expected value of the total score x = P i x i is E ( x ) = X i E ( x i ) = np. This is the expected value of the binomial distribution. To find the variance of the Bernoulli trial, we use the formula E ( x ) = E ( x 2 ) { E ( x ) } 2 . For a single trial, there is E ( x 2 i ) = X x i =0 , 1 f ( x i ) = p, V ( x i ) = p p 2 = p (1 p ) = pq, where q = 1 p.
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  • Spring '12
  • D.S.G.Pollock
  • Normal Distribution, Probability theory, probability density function, dx, Ø

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