# Warm up questions

• 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?

# Stochastic domination and the inspection paradox

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.

# Baby wald

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 $\mathbb{E} \big( \sum_{i=1} ^N X_i \big) = \mathbb{E}N \mathbb{E} X_1$

# Tiling

• 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.