How could know the average salary of your friends 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?
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 I tie, where I win; what is the probability that I will win?
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 Swift’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.