Analyze distribution of a 3-dice product

You roll three independent fair 10-sided dice. Each die shows an integer in $\{0,1,\dots,9\}$ with equal probability. Let the outcomes be $X,Y,Z$, and define the product $P = XYZ$.

1. Mean : Compute $\mathbb{E}[P]$ .
2. Median inference :
   • (a) Find the population median of $P$ (i.e., the smallest $m$ such that $\Pr(P\le m)\ge 0.5$ ).
   • (b) Suppose you roll the three dice $n$ times i.i.d. and compute the sample median $\tilde P$ . Describe how you would construct an approximate 90% confidence interval for the population median using $\tilde P$ . (A rough numerical interval is OK if you make clear what $n$ you are assuming.)
3. Mode and skew :
   • (a) What is the mode of $P$ ?
   • (b) How does the mode/skewness inform how you would place or adjust the interval from (2b)?
4. Market-making (interactive) : You must quote a bid/ask market of width 5 for the (unknown-to-the-interviewer) population median $m$ of $P$ : you quote a bid $b$ and ask $a$ with $a-b=5$ . The interviewer may trade 1 unit each round:
   • If they buy , you sell 1 unit at price $a$ (you become short).
   • If they sell , you buy 1 unit at price $b$ (you become long).
   After 4 rounds, the position is settled at the true median $m$ (cash-settled), so P&L for each unit is (sell price $-m$ ) if you sold, or ( $m$ -buy price) if you bought.
   • (a) Propose an initial market $[b,a]$ .
   • (b) Explain how you would update $[b,a]$ after each trade to incorporate the information revealed.
   • (c) Show how to compute the resulting P&L given any sequence of 4 buys/sells.