In another post I discussed how the metaphor of baking helps me retain intuition about the all-important statistical concept of variance, insofar as it draws a very clear line between what I can know theoretically, versus what I can only know empirically, that is, through taking samples.

This post will extend the metaphor of statistics as baking with math, to build intuition about the expected value and variance of composite random variables. Baking lends itself well to the idea of composite random variables, because it's easy to see how some ingredients might be well incorporated into one and the same batter, whereas other ingredients might be

Composite random variables operate the same way as simple random variables, except there are more than one. In the last post, I talked about the variance being a measure of *how well mixed* is a proverbial cake batter. Poorly mixed batter might have huge clumps of flour in one spoonful, and almost no flour in another. But not all ingredients in the batter are added at the same time. Batter may be very well mixed with respect to sugar, but not well mixed with respect to flour.

Take, for example, a recipe in which you mix sugar, oil, eggs, and vanilla. Then you add flour, only after the other ingredients are mixed. Obviously, you should mix all of your ingredients well, but maybe you got a pressing phone call, or one of your kids needed help on the potty. So you're in a hurry when you add your flour, and even though you did a really good mixing job with the first round of ingredients, the flour is not well incorporated.

As was the case with the single ingredient, the "expected value" of any ingredient is known at the outset. If I follow a recipe, I know the expected amount of any of the ingredients before I can even mix them. Why? Because it's right there in the ingredients list! If my recipe is 60% flour and 40% water, then I know that E(Flour)=.6*(Units Flour) and E(Water) is .4*(Units Water). However, if you ask me how likely it is that any spoonful of the batter will contain 60/40 flour/water, I cannot answer this question unless I know how well mixed the batter is, *with respect to each ingredient*.