The discussion revolves around the application of the law of iterated expectation in probability derivations, specifically in the context of calculating the expected number of flips required to achieve a certain number of consecutive heads in a coin flipping scenario.
The discussion does not reach a consensus, as participants are still seeking clarification and understanding of the application of the law of iterated expectation. There are multiple viewpoints regarding the derivation and its implications.
Participants have not fully defined the variables involved, and there is some ambiguity regarding the transition from the conditional expectation to the expected value. The discussion reflects varying levels of understanding among participants.
Sorry, I realize I didn't explain this well. Rather than retype everything, and since the website is very clear, perhaps the link is easier? It's here. I'm wondering how they applied the law of iterated expectation to arrive from equation 3 to 4.anuttarasammyak said:Could you tell me more detail on the setting ? What are E, X_n and X_n|Xn-1 ?
Wow, I feel like a moron. Can we just say I was exhausted and that's why I was confused? Sheesh...thanks though!pbuk said:We are given ## E(X_n | X_{n-1}) = X_{n-1} + f ##.
Take the expected value of both sides: ## E \left [ E(X_n | X_{n-1}) \right ] = E \left [ X_{n-1} + f \right ] ##.
From the law of iterated expectation we have ## E \left [ E(X_n | X_{n-1}) \right ] = E \left [ X_n \right ] ##.