[Statistics 110] Probability

Mosteller-Wallace, The Federalist PapersGoverment : IQSSFinace : STAT 123Gambling : Fermat-Pascal 1650'sLife : Statistics is the logic of uncertaintyA

Tips for homework Labling people, object, etc. -> 1, 2, ... n : 10 people, split into team of 6, team of 4 => $\begin{pmatrix}10\\4\\ \end{pmatrix}

In k people, let's find the probability that 2 have same birthdays.Just for simplicity, exclude Feb 29th; assume other 265 days equally likely; assume

$Aj$ : jth card in desk is labeled; Find $P(\\displaystyle ⋃^{n}{j=1} A_j)$$P(A_1 \\cup A_2 \\cup\\; ... \\; \\cup A_n)$, there are $\\begin{pmatrix}n

Thinking conditionally is condition for thinking!How to solve a problemTry simple and extreme casesBreak up problem into simpler pieces.: Let $A_1 \\;

Monty Hall : the game show hostThere are three doors, Monty hall ask to choose. (Monty knows which)1 door has car, 2 doors have goatsMonty always open

Binomial distribution Bin(n, p) $X ~ Bin(n, p)$ Story: X is \# of successes in n independent Bern(p) trials Bern(p) p: prob. success Sum of indic

CDF(누적 분포 함수) : $F(x) = P(X \\le x)$, as a function of real x.Find $P(1 < X \\le 3)$ using F: $P(X \\le 1) + P(1 < X \\le 3) = P(X \\le 3)$$\\Ri

Let $T = X + Y$, show $E(T) = E(X) + E(Y)$$\\displaystyle \\sum{t} t P(T=t) \\; ?= \\displaystyle \\sum{x} x P(X=x) + \\displaystyle \\sum\_{y} y P(Y=

Don't confuse a r.v. with its distributionSum of r.v. vs Sum of PMF is different!"Word is not the thing, the map is not the territory."random variable

Defn R.V $X$ has PDF $f(x)$ if $P(a \\le X \\le b) = \\int\_{a}^{b} f(x)dx$For all a, b $a=b \\Rightarrow \\int\_{a}^{a} f(x)dx = 0$To be valid, $f(x)

Let F be a continuous, strictly increasing CDFThen $X=F^{-1} \\sim F$ if $U~Unif(0, 1)$It used in Diffusion model ans Simulating some distributionAlso

$Z \\sim N(0, 1)$CDF $\\Phi$$E(Z) = 0$$Var(Z) = E(Z^2) = 1$$E(Z^3) = 0$$\\int\_{-∞}^{∞} z^3 \\displaystyle \\frac{1}{\\sqrt{2 \\pi}} e^{-z^2/2} dz$ (o

n toy types, equally likeyly, Find expected time $T$ (i.e. \\ n 가지 장난감을 모아야 전체를 모은다고 할 때, 장난감 전부를 모으는 데까지 걸리는 시간 T(뽑아야 하는 장난감 수)의 기댓값을 구하시오. $T = T_1

rate(속도) parameter $\\lambda$$X \\sim Expo(\\lambda)$ has PDF: $\\lambda e^{-\\lambda x}$, $x > 0$ (0 otherwise)valid $\\int\_{0}^{∞} \\lambda e^{-\\l

Intuition of Memoryless Property $E(T | T > 20) > E(T)$ 20대 이상의 기대 수명(T)은 무작위적인 기대 수명에 비해 큰 것이 사실이다. 모든 사람의 기대 수명이 같지 않고 다 가변성이 있다면 이는 의미가 있다. If

$X \\sim Expo(1)$, find MGF, moments cf. 물리학에서의 관성 모멘트와 분산이 관련이 있음MGF: $M(t) = E(e^{tx}) = \\int{0}^{∞} e^{tx} e^{-x} dx = \\int{0}^{∞} e^{-x(1-t)} dx

joint CDF $F(x, y) = P(X \\le x, Y \\le y)$joint PDFcontinuous case: $f(x, y) = \\displaystyle \\frac{\\partial}{\\partial x \\partial y} F(x, y)$$P((

Distance between 2 i.i.d. variablesEx) Find $E|Z_1 - Z_2|$, with $Z_1$, $Z_2$ $\\sim i.i.d. N(0, 1)$$Z_1 - Z_2$ : i.i.d. standard normalThm) $X \\sim

Defn. : need 2 variables$Cov(X, Y) = E((X-EX)(Y-EY)) = E(XY) - E(X)E(Y)$If X is bigger than EX, Y is also bigger than EY.If X is smaller than EX, Y is

Variance of Hypergeom(w, b, n)$p = \\displaystyle \\frac{w}{w+b}$, $w+b = N$$Var(\\displaystyle \\sum{j=1}^{n} X_j) = Var(X_1) + ... + Var(X_n) + 2 \\

$Beta(a, b)$, $a>0, b>0$PDF: $f(x) = cx^{a-1}(1-x)^{b-1}$, $0 < x < 1$Features"Flexible" family of continuous distributions on $(0, 1)$(1) if $a