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Unformatted text preview: Application Example 15 (Conditional second-moment analysis) UNCERTAINTY UPDATING USING NOISY OBSERVATIONS One of the uses of conditional distributions is in updating uncertainty on a variable of interest X based on observation of one or more other variables. For example, one may want to update uncertainty on rainfall tomorrow based on observation of rainfall today, the strength of beam 1 based on observation of the strength of beam 2, or soil compressibility at location A given soil compressibility at some other location B. In certain cases, the observed variable is itself a measurement of X. For example, one may measure the strength of a concrete column by some nondestructive test, measure topographic elevation at a point using a satellite instrument with limited accuracy, or sample the water of a stream with an imprecise device to determine its degree of contamination. In all these cases, the measurement is not exact. We want to see how, based on such noisy data, one can update uncertainty on the quantity of interest X. The method described below is exact if the random variables involved are normally distributed, but is often used as an approximation for variables with any distribution. Conditional Distributions of Variables with Joint Normal Distribution Let X 1 and X 2 be jointly normal variables with mean values m 1 and m 2 , variances 1 2 and 2 2 , and correlation coefficient . 0ne can show that the conditional distribution of (X 1 |X 2 =x 2 ) is also normal, with mean value m 1|2 and variance 1|2 2 given by m 1|2 = m 1 + 1 ( x 2 m 2 ) 2 (1) 2 = 1 2 (1 2 ) 1|2 1 Notice that the conditional mean depends on the observed value x 2 of X 2 , whereas the conditional variance does not. Moreover, the conditional variance differs from the unconditional variance by the factor (1 - 2 ), which is smaller than 1 whenever X...
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- Fall '07