In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.
E [ f ( X ) ] = E [ f ( μ X + ( X − μ X ) ) ] ≈ E [ f ( μ X ) + f ′ ( μ X ) ( X − μ X ) + 1 2 f ″ ( μ X ) ( X − μ X ) 2 ] . Notice that E [ X − μ X ] = 0 , the 2nd term disappears. Also E [ ( X − μ X ) 2 ] is σ X 2 . Therefore,
E [ f ( X ) ] ≈ f ( μ X ) + f ″ ( μ X ) 2 σ X 2 where μ X and σ X 2 are the mean and variance of X respectively.
It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions. For example,
E [ X Y ] ≈ E [ X ] E [ Y ] − cov [ X , Y ] E [ Y ] 2 + E [ X ] E [ Y ] 3 var [ Y ] Analogously,
var [ f ( X ) ] ≈ ( f ′ ( E [ X ] ) ) 2 var [ X ] = ( f ′ ( μ X ) ) 2 σ X 2 . The above is using a first order approximation unlike for the method used in estimating the first moment. It will be a poor approximation in cases where f ( X ) is highly non-linear. This is a special case of the delta method. For example,
var [ X Y ] ≈ var [ X ] E [ Y ] 2 − 2 E [ X ] E [ Y ] 3 cov [ X , Y ] + E [ X ] 2 E [ Y ] 4 var [ Y ] . 