∙We will often be interested inEY2|Xx.∙As we saw in the example of an exponential distribution, if weconstruct conditional distributions from the unconditional distributionswe already know, then finding conditional moments is trivial.∙In many applications, the main feature that we must specifyistheconditional mean. Often fairly simple forms are used, such asEY|XxxEY|Xxxx2and so on.30
∙IfXis a random vector, linearity is often assumed:EY|Xx1x1...kxk≡xwherexx1,x2,...,xkis a row vector.∙OftenXcontains nonlinear functions of underlying variables. In EC820B, you will see that it is linearity in the parametersandthat hasconsequences for estimation.31
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∙Generally, if we writeEY|Xxxthen we are interested inknowing howxchanges as the elements ofxchange. Often, at leastfor continuous elements, we focuse on the partial derivatives,∂x∂xj.For discrete changes, we can look at differences evaluated at two valuesofx.32