COMP 2711H: Lecture 34

Date: 2024-11-18 17:57:54

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❓Questions

Notes

Problems

Problem 1: expectation of the sum
- toss two fair dice. what is the expected value of the sum?
- each pair: w/ same probability
- you can manually compute, but...
- $X_{i}$ : no. on the $i$ 0th die
  - $X = X_{1} + X_{2}$
  - $E [X] = E [X_{1}] + E [X_{2}] = 3.5 + 3.5 = 7$ (no independence needed)
Problem 2: no. of heads
- w/ biased coin with head on $p$ chance
- what is expectation of heads after flipping it $n$ times?
- $E [X] = \sum_{i = 0}^{n} i \cdot P (X = i)$
  - $P (X = i) = (i n) p^{i} (1 - p)^{n - 1}$
  - then $E [X] = \sum_{i = 0}^{n} i \cdot (i n) p^{i} (1 - p)^{n - 1}$
  - 👨‍🏫: correct, but ugly!
- simply: use indicator r.v. $X_{i} = {10 if i -th flip is H if i -th flip is T$
- $X = \sum_{i = 1}^{n} X_{i}$
- $E [X] = \sum_{i = 1}^{n} E [X_{i}] = \sum_{i = 1}^{n} P [X_{i} = 1] = \sum_{i = 1}^{n} p = n p$
- 👨‍🏫 lesson: life is easier w/ linearity of expectation
Problem 3: fair coin toss
- a game w/ fair coin
- keep flipping until you get $H$
- $X =$ no. of flips
- 👨‍🏫: what is $E [X]$ ?
  - intuitively: 2
  - proof: let $μ := E [X]$ $μ μ = \frac{1}{2} \cdot 1 + \frac{1}{2} (1 + μ) = 2$
Problem 4: biased coin toss until...
- w/ $p$ chance of success: $μ μ = p \cdot 1 + (1 - p) (1 + μ) = \frac{1}{p}$
Problem 5: hat-trade
- $n$ people in party, with a hat
- each person $i$ : leaves w/ hat $π [i]$
- what is: expected no. fo people w/ their own hat?
  - $∣ {i ∣ i = π [i]} ∣$
- $E [X] = \sum_{i = 0}^{n} i \cdot P [X = i]$
- $P [X = i] = \frac{( i n ) D _{n - 1}}{n !}$
  - 👨‍🏫 hard to compute!
- let $X$ : sum of i.r.v. $X_{i} = {10 if i = π [i] otherwise$
- $X = \sum_{i = 1}^{n} X_{i} ⟹ E [X] = \sum_{i = 1}^{n} E [X_{i}]$
  - $E [X_{i}] = P [X_{i} = 1] = \frac{( n - 1 )!}{n !} = \frac{1}{n}$
  - $X = \sum_{i = 1}^{n} \frac{1}{n} = 1$
  - 👨‍🏫 so simple!
Problem 6 (IMO 1987)
- let $p_{n} (k)$ be no. of permutations of $[n]$ w/ exactly $k$ fixed points
  - $n \geq 1$
- prove
  - $\sum_{k = 0}^{n} k \cdot p_{n} (k) = n!$
  - divide both sides by $n!$ , and it's a probabilities problem!
    - $\sum_{k = 0}^{n} k \cdot \frac{p _{n} ( k )}{n !} = 1$
    - $E [X] = \sum_{k = 1}^{n} k \cdot P (X = k) = 1$ , where $X$ : no. of fixed points
      - shown from problem 1
Problem 7: coupon collector
- $n$ types of coupons, buying a coupon every day
  - type: u.a.r.
- let $X$ : no. of days of buying coupons
  - what is $E [X]$ ?
- finding it from $E [X] = \sum_{i = 1}^{\infty} i \cdot P [X = i]$
  - you can do so with
    - $P [X = i] = \frac{no. of onto functions ( i - 1 , n - 1 )}{n !}$
    - 👨‍🏫 but painful...
- let $X_{i}$ : no. of steps to get from $i$ -th to $i + 1$ -th new type
  - let $X = \sum_{i = 1}^{n} X_{i}$
  - $E [X_{0}] = 1$ (constant)
  - $E [X_{1}] = \frac{n}{n - 1}$ as:
    - chance of getting a new type: $\frac{n - 1}{n}$
  - $E [X_{i}] = \frac{n}{n - i}$ as:
    - chance of getting a new type: $\frac{n - i}{n}$
- $E [X] = \sum_{i = 0}^{n} E [X_{i}] = n \cdot \sum_{i = 0}^{n - 1} \frac{1}{n - i} = n \cdot \sum_{i = 1}^{n} \frac{1}{i} = n \cdot H_{n}$
  - takes $n lo g n$ time!
Problem 8: USA MTS
- mathematical talent sets
- play a game: pot starting at $0 - e v ery t im e : f l i p a f ai rco in - i f i t^{'} s$ H $: a dd$ 100 to the pot
- if it's tails: pot becomes 0 and get a strike
  - 3 strikes: loss
- before each flip: you can claim the pot
- suppose optimal play, what is expected value of earnings?
- $E [X, c] = \frac{1}{2} (E [X, c] + 100) + \frac{1}{2} (E [X, c - 1])$
  - $c = 2$ initially (no. of chances remaining)
  - $E [X, 0] = \frac{1}{2} (100 + E [X, 0])$
    - $E [X, 0] = 100$ $E [X, 2] \frac{1}{2} E [X, 2] E [X, 1] \frac{1}{2} E [X, 1] E [X, 2] = \frac{1}{2} (E [X, 2] + 100) + \frac{1}{2} (E [X, 1]) = 50 + \frac{1}{2} (E [X, 1]) = \frac{1}{2} (E [X, 1] + 100) + \frac{1}{2} (E [X, 0]) = 50 + 50 = 100 = 2 \times 150 = 300$
- flip the coin: if expected value of earning > money in the pot

COMP 2711H: Honors Discrete Mathematical Tools for Computer Science