Undergrad Understanding the Law of Iterated Expectation in Probability Derivations

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SUMMARY

The discussion centers on 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 n consecutive heads in a coin toss scenario. The key equations presented are E(X_n | X_{n-1}) = X_{n-1} + f and E(X_n) = E(X_{n-1}) + f, illustrating how the law facilitates the transition from conditional expectations to unconditional expectations. The participants clarify the definitions of E, X_n, and X_{n-1}, emphasizing the importance of understanding these terms for proper application of the law.

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I'm reading a website where they're doing a derivation. Within the derivation they write $$E(X_n | X_{n-1}) = X_{n-1} + f \implies E(X_n) = E(X_{n-1} ) + f$$. Evidently the implication stems from the law of iterated expectation, but I can't see how. If it helps, the question asked is "what is the expected number of flips for a coin to achieve ##n## consecutive heads.
 
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Could you tell me more detail on the setting ? What are E, X_n and X_n|Xn-1 ?
 
anuttarasammyak said:
Could you tell me more detail on the setting ? What are E, X_n and X_n|Xn-1 ?
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.
 
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 ] ##.
 
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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 ] ##.
Wow, I feel like a moron. Can we just say I was exhausted and that's why I was confused? Sheesh...thanks though!
 

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