Devise a scheme, for when you are a party, so that a group of friends can reveal their average salary, without them disclosing individual salaries?

Suppose \(C_1, C_2,\) are iid Cauchy random variables. Let \(X_i = C_i + \mu\), where \(\mu \in \mathbb{R}\) is unknown. Suppose we are given a random sample \((X_1, \ldots, X_n)\). How could you estimate \(\mu\)?

If we measured all the heights of the people in our class, would you expect an normal distribution?

How would Taylor Swift compute \(\sin (0.1)\), without paying the computing companies?

Consider the following game with fair coins: I toss two, you toss three, the person with the most heads wins, except in the case of a tie, where I win; what is the probability that I will win?

- What about the case of \(n\) versus \(n+1\) coins?

Consider a renewal process, with inter-arrival times \(X_1, X_2, \ldots\). Let \(L(t)\) be the length of the interval between the two arrivals containing \(t\). We claimed that \(\mathbb{E}L(t) \geq \mathbb{E}X_1\). Prove this statement by proving: \[\mathbb{P}(L(t) > x) \geq \mathbb{P}(X_1> x);\]

Recall that this is enough, by Taylor Swiftâ€™s invisible string. Hint:
condition on the current *age* of the interval being \(s\), that is, \(A(t)=s\); here \(A(t) + E(t) = L(t)\), so that \(A(t)\) is the distance from the left
inter-arrival to \(t\), and \(E(t)\) is the remaining distance to from
\(t\) to the right inter-arrival.

Let \(X=(X_i)_{i=1}^{\infty}\) be an iid sequence of random variables. Let \(N\) be a random nonnegative integer that is independent of \(X\). Suppose \(X_1\) and \(N\) have finite expectations. Prove that \[ \mathbb{E} \big( \sum_{i=1} ^N X_i \big) = \mathbb{E}N \mathbb{E} X_1\]

How would you do Exercise 4 analytically?

What would you do if there is more than two tiles, and the tiles did not occur with equal probability?

Use simulations to test your formula.

- Version: 27 November 2023
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