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approximate the expectation of almost arbitrary functions of a random
variable X.
BytheDeltaMethod,theapproximateexpectationandvarianceofarandom
variable X is given as follows:
Where f”(a) is the second derivative of the function f(x). This approximation
derives from the use of the Taylor Expansion, which is often used in applied
mathematics to approximate complicated non-linear functions. The Taylor
Expansion provides an expression for expanding a function into several
parts when evaluated with respect to a point “a.” For example, this is the
Taylor Expansion for f(x) containing the linear and quadratic parts of the
expansion, known as the second order expansion:
The remaining parts of the Taylor Expansion are captured by the error term
“e,” which contains the cubic and higher order expansions. This can be used
to compute the error of the approximation, but for the purposes of this topic
that error will be considered to be “small enough.”
To derive the Delta Method results, the function of the random variable X,
f(X), is expanded around its own expectation E[X], giving the formula:
Taking the expectation of this expansion and solving yields the second order
approximation of E[f(X)]:
In this case, E[X] is a constant value, so f(E[X]) is also a constant. The
expectation of a constant is the constant, E[b] = b, and the expectation of b
