A more accurate approximation to E[h(X1, . . . , Xn)] in Exercise 93 is
Compute this for Y = h(X1, X2, X3, X4) given in Exercise 93, and compare it to the leading term h(μ1, . . . , μn).
Reference Exercise 93
Let X1, . . . , Xn be independent rv’s with mean values μ1, . . . , μn and variances . Consider a function h(x1, . . . , xn), and use it to define a new rv Y = h(X1, . . . , Xn). Under rather general conditions on the h function, if the σi’s are all small relative to the corresponding μi’s, it can be shown that where each partial derivative is evaluated at (x1, . . . , xn) = (μ1, . . . , μn). Suppose three resistors with resistances X1, X2, X3 are connected in parallel across a battery with voltage X4. Then
by Ohm’s law, the current is
Let μ1 =10 ohms, σ1 1.0 ohm, μ2 = 15 ohms, σ2 = 1.0 ohm, μ3 = 20 ohms, σ3 = 1.5 ohms, μ4 =120 V, σ4 = 4.0 V. Calculate the approximate expected value and standard deviation of the current (suggested by “Random Samplings,”CHEMTECH, 1984: 696–697).
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