17: Probability - Exp, Var, Inequalities

1) Expectation & Variance Basics

01 Modified Die: Probability and Moments

Vahe added a dot on the 4 side of the die, making it 5, and then added two dots on the 1 side, making it 3.

1. What is the probability that the outcome of the die is greater than 4?
1. Find the expectation and variance of the modified die.

02 Die Game: Expected Value

You roll a fair die. If you roll 1, you are paid $25. If you roll 2, you are paid $5. If you roll 3, you win nothing. If you roll 4 or 5, you must pay $10, and if you roll 6, you must pay $15.

1. Compute the expected payoff.
1. Do you want to play?

03 Uniform Sum Expectation

Let $X$ and $Y$ be two continuous random variables with uniform distribution on $(0,2)$.

Find $\mathbb{E}[X+Y]$.

2) LOTUS: Law of the Unconscious Statistician

04 Expectation Without the CDF

Let $X\sim\mathrm{Uniform}(0,1)$. Define $Y=\log(1+X)$.

1. Compute $\mathbb{E}[Y]$ using LOTUS directly.
1. Compute $\operatorname{Var}(Y)$ (you may leave integrals in closed form).

05 Piecewise Payoff

Let $X\sim\mathrm{Exp}(\lambda)$. A “refund policy” pays $g(X)=\min(X,c)$ for fixed $c>0$.

(for $\mathrm{Exp}(\lambda)$, $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ and $F(x)=1-e^{-\lambda x}$ for $x\ge 0$. Aprikyan will cover distibutions during next or next-next lesson.)

1. Compute $\mathbb{E}[g(X)]$ using LOTUS.
1. Compute $\mathbb{P}(g(X)=c)$.
1. Find $\dfrac{d}{dc}\,\mathbb{E}[g(X)]$ and interpret.

3) Expectation & Variance (Decision / Games)

06 When to Stop (Secretary-lite)

You see prices of used laptops one by one, i.i.d. $\mathrm{Uniform}(0,1)$. You can accept one price and stop, or reject and continue; once rejected, it’s gone. You must decide a stopping rule.

1. Consider the rule: “accept the first price $\le t$.” Compute the expected accepted price as a function of $t$ given a maximum of $N$ offers.
1. Find (approximately) the best $t$ for $N=10$.

07 Optimal Reroll (Single Reroll Allowed)

You roll a die once; you may choose to keep it or reroll once (then must keep). Goal: maximize expected value.

1. What threshold rule is optimal?
1. What is the resulting expected value?
1. Compute the variance of the final payoff under the optimal strategy.

08 St. Petersburg Game (Bonus)

A fair coin is tossed until the first Heads appears. If Heads appears on toss $k$, you get $2^k$ dollars.

1. Compute the expected payoff.
1. Why might people still refuse to pay an “infinite fair price” to play?

4) Indicators & Counting (Expectations via Linearity)

09 Coupon Collector: Sticker Packs

A shop gives one random sticker from a set of $n$ stickers with each purchase (uniform, independent).

1. Expected number of purchases to collect all $n$ stickers (you may express the answer using harmonic numbers).
1. For $n=50$, give a rough numerical approximation.

10 Distinct Stickers After $n$ Packs: $\mathbb{E}[D]$, $\operatorname{Var}(D)$

You buy $n$ sticker packs; each pack contains one sticker uniformly from $\{1,\dots,m\}$, independent. Let $D$ be the number of distinct stickers you have after $n$ packs.

1. Find $\mathbb{E}[D]$.
1. Find $\operatorname{Var}(D)$ using indicators and pairwise terms.

5) Inequalities (Markov & Chebyshev)

11 Markov — “What’s the chance my bill is huge?”

Your monthly electricity bill $B\ge 0$ has average $\mathbb{E}[B]=\$80$.

1. Use Markov’s inequality to bound $\mathbb{P}(B\ge \$200)$ and $\mathbb{P}(B\ge \$300)$.
1. Suppose the provider claims: “the probability of a $\$300+$ bill is at most 5%.” What average bill $\mathbb{E}[B]$ would make this statement true by Markov?

12 Chebyshev — “Commute-time reliability”

Commute time $T$ (minutes) has mean $\mu=40$ and variance $\sigma^2=25$ (so $\sigma=5$).

1. Use Chebyshev’s inequality to upper-bound $\mathbb{P}(T\ge 55)$ and $\mathbb{P}(T\le 25)$.
1. How large must a time buffer $b$ be so that $\mathbb{P}(T\le \mu+b)\ge 0.95$?

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--- title: "17: Probability - Exp, Var, Inequalities" format: html: css: homework-styles.css --- <script src="homework-scripts.js"></script> --- ## 1) Expectation & Variance Basics ### 01 Modified Die: Probability and Moments {data-difficulty="1"} Vahe added a dot on the 4 side of the die, making it 5, and then added two dots on the 1 side, making it 3. - a) What is the probability that the outcome of the die is greater than 4? - b) Find the expectation and variance of the modified die. ### 02 Die Game: Expected Value {data-difficulty="1"} You roll a fair die. If you roll 1, you are paid $25. If you roll 2, you are paid $5. If you roll 3, you win nothing. If you roll 4 or 5, you must pay $10, and if you roll 6, you must pay $15. - a) Compute the expected payoff. - b) Do you want to play? ### 03 Uniform Sum Expectation {data-difficulty="3"} Let $X$ and $Y$ be two continuous random variables with uniform distribution on $(0,2)$. - Find $\mathbb{E}[X+Y]$. --- ## 2) LOTUS: Law of the Unconscious Statistician ### 04 Expectation Without the CDF {data-difficulty="2"} Let $X\sim\mathrm{Uniform}(0,1)$. Define $Y=\log(1+X)$. - a) Compute $\mathbb{E}[Y]$ using LOTUS directly. - b) Compute $\operatorname{Var}(Y)$ (you may leave integrals in closed form). ### 05 Piecewise Payoff {data-difficulty="2"} Let $X\sim\mathrm{Exp}(\lambda)$. A “refund policy” pays $g(X)=\min(X,c)$ for fixed $c>0$. (for $\mathrm{Exp}(\lambda)$, $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ and $F(x)=1-e^{-\lambda x}$ for $x\ge 0$. Aprikyan will cover distibutions during next or next-next lesson.) - a) Compute $\mathbb{E}[g(X)]$ using LOTUS. - b) Compute $\mathbb{P}(g(X)=c)$. - c) Find $\dfrac{d}{dc}\,\mathbb{E}[g(X)]$ and interpret. ## 3) Expectation & Variance (Decision / Games) ### 06 When to Stop (Secretary-lite) {data-difficulty="3"} You see prices of used laptops one by one, i.i.d. $\mathrm{Uniform}(0,1)$. You can accept one price and stop, or reject and continue; once rejected, it’s gone. You must decide a stopping rule. - a) Consider the rule: “accept the first price $\le t$.” Compute the expected accepted price as a function of $t$ given a maximum of $N$ offers. - b) Find (approximately) the best $t$ for $N=10$. ### 07 Optimal Reroll (Single Reroll Allowed) {data-difficulty="2"} You roll a die once; you may choose to keep it or reroll once (then must keep). Goal: maximize expected value. - a) What threshold rule is optimal? - b) What is the resulting expected value? - c) Compute the variance of the final payoff under the optimal strategy. ### 08 St. Petersburg Game (Bonus) {.bonus-problem data-difficulty="3"} A fair coin is tossed until the first Heads appears. If Heads appears on toss $k$, you get $2^k$ dollars. - a) Compute the expected payoff. - b) Why might people still refuse to pay an “infinite fair price” to play? --- ## 4) Indicators & Counting (Expectations via Linearity) ### 09 Coupon Collector: Sticker Packs {data-difficulty="3"} A shop gives one random sticker from a set of $n$ stickers with each purchase (uniform, independent). - a) Expected number of purchases to collect all $n$ stickers (you may express the answer using harmonic numbers). - b) For $n=50$, give a rough numerical approximation. ### 10 Distinct Stickers After $n$ Packs: $\mathbb{E}[D]$, $\operatorname{Var}(D)$ {data-difficulty="3"} You buy $n$ sticker packs; each pack contains one sticker uniformly from $\{1,\dots,m\}$, independent. Let $D$ be the number of distinct stickers you have after $n$ packs. - a) Find $\mathbb{E}[D]$. - b) Find $\operatorname{Var}(D)$ using indicators and pairwise terms. --- ## 5) Inequalities (Markov & Chebyshev) ### 11 Markov — “What’s the chance my bill is huge?” {data-difficulty="2"} Your monthly electricity bill $B\ge 0$ has average $\mathbb{E}[B]=\$80$. - a) Use Markov’s inequality to bound $\mathbb{P}(B\ge \$200)$ and $\mathbb{P}(B\ge \$300)$. - b) Suppose the provider claims: “the probability of a $\$300+$ bill is at most 5%.” What average bill $\mathbb{E}[B]$ would make this statement true by Markov? ### 12 Chebyshev — “Commute-time reliability” {data-difficulty="2"} Commute time $T$ (minutes) has mean $\mu=40$ and variance $\sigma^2=25$ (so $\sigma=5$). - a) Use Chebyshev’s inequality to upper-bound $\mathbb{P}(T\ge 55)$ and $\mathbb{P}(T\le 25)$. - b) How large must a time buffer $b$ be so that $\mathbb{P}(T\le \mu+b)\ge 0.95$? # 🎲 xx+37 (xx) - ▶️[ToDo]() - 🔗[Random link ToDo]() - 🇦🇲🎶[ToDo]() - 🌐🎶[ToDo]() - 🤌[Կարգին ToDo]() <a href="http://s01.flagcounter.com/more/1oO"><img src="https://s01.flagcounter.com/count2/1oO/bg_FFFFFF/txt_000000/border_CCCCCC/columns_2/maxflags_10/viewers_0/labels_0/pageviews_1/flags_0/percent_0/" alt="Flag Counter"></a>