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How does it develop the notion of a martingale
just taking expectation.
each to a dimension in . For example, if our probability space has two elements, (un)conditional expectation is orthogonal projection onto the line y = x. Here, y = x encodes the fact that with the unconditional expectation, our estimate of *Vintage Rolex Watches For Sale Uk* X has to be constant on all possible realizations/omegas.
is 100% valid. If X is merely in , then we are forced to define conditional expectation as a limit of projections of truncated versions of X which are square integrable. So, the intuition slightly breaks down, but in a majority of applications, you are able to work with square integrable random variables, and the projection intuition *Gold Rolex Watch Price List*
*Geometrically, you can think of conditional expectation as a projection (more on this at the end). Suppose X is a random variable known at time t = 2, and we condition it on the information known at t = 1. By taking conditional expectation with respect to time 1 information, we project X from the space of 2 measurable functions onto the subspace of 1 measurable functions.
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*I just attempt to answer your last question, why is expectation of a conditional expectation the same as **Rolex Submariner No Date Nato*
*For intuition purposes, think of X as a point living in 3 d space. By taking conditional expectations, we project it into 2 d space. By taking unconditional expectation, we project it onto 1 d space (getting a single number in return). But if you think about projections geometrically, we know that projecting it into 2 d and then 1 d is the same as just projecting it to 1 d in one fell swoop. Draw this picture in your mind to convince yourself. Doing the projection in two steps is like taking expectations of the conditional expectation, and projecting it in one step is like just taking unconditional expectations.
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*When X is a discrete random variable, we can make this description rigorous, by assigning **Rolex Daytona Leather Strap*
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