Toss two coins, and , independently of one another, with probabilities respectively of giving heads. What is the probability

…but the coins are independent! Explain.

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Toss two coins, and , independently of one another, with probabilities respectively of giving heads. What is the probability

…but the coins are independent! Explain.

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Construct a complex-analytic function on the open unit disk such that all derivatives admit continuous extensions to the closed disk , but such that cannot be analytically continued to any strictly larger domain .

I mentioned Sanov’s theorem in a previous post. This is a fun variant on the same argument.

Let be a Geometric distribution on given by: , so that the mean is , and let be independent and identical samples from this distribution. For , let be the empirical distribution of , that is,

Construct explicitly probability measures and such that, for all ,

If you know about entropy of probability measures, show also that these changes of measure have asymptotically finite entropy:

*There are more counterexamples in heaven and earth than are dreamt of in your real analysis course* – Hamlet, deleted scene.

Show that there exists a smooth function such that, for all , the Taylor series has zero radius of convergence.

*Note that this is different from* this previous problem*.Why?*

*Bonus points for an explicit construction.*

Let be a compactly supported probability measure on . For , let be the Gaussian distribution of variance .

Fix a continuous function , and for , define a local average

Suppose that has a density . Making any assumptions necessary on the regularity of , show that converges to a limit for all . Must be continuous?

What happens if we work only with a compactly supported probability measure ?

You are offered a choice between two envelopes: envelope A contains , while envelope B contains with probabilitity , and is otherwise empty, with . With constant average risk aversion , which option do you prefer?

A prankster wishes to waste your time by making you make a large number of choices, while still making the same amount of money available for prizes. You will be given a large number choices between envelopes A, which contain , or envelopes B, which contain with probability , independently of each other. What is your strategy in this case? Do you complain about this prank?

Does your answer change if you are allowed to open the envelopes before making the next choice, or if you only open all of them at the end?

What is your optimal strategy if you are instead told that exactly of the B envelopes contain the prize, but you do not find out about those you do not open?

Consider the *antiferromagnetic Ising Model* on a finite graph , where each site is assigned a spin , and the probability of a given configuration is , , so that nearest neighbours are encouraged to have *opposite* spins.

What do typical configurations look like if is a (finite) square lattice in the limit ? What about if is a triangular lattice?

Show that the space of smooth functions admits a complete metric such that if and only if the derivatives converge uniformly to for all .

Show that the set of functions which are given by a power series about any point is meagre in the sense of Baire.

Construct a continuous local martingale with , such that for all .

Classify all Banach spaces such that there is a surjection , which is continuous when is equipped with, respectively, the norm topology or the weak topology.

What about if is no longer Banach?

*This is a more high-level version of a previous post. Solving the previous problem may make this problem easier and/or less interesting.*