Title: Probability
Tags: probability

Q: Bayes' theorem
A: routes to A via B out of all routes to A

Q: variance
A: the spread around the center;
weighted average of squared distance from mean

standard deviation

Q: sampling formula for replacement:[...] ordered:[...]
=
[...]
A: YES; YES; \(n^k\)

Q: sampling formula for replacement:[...] ordered:[...]
=
[...]
A: YES; NO; \(\binom{n+k-1}{k}\)

Q: sampling formula for replacement:[...] ordered:[...]
=
[...]
A: NO; YES; \(n(n-1)\cdots(n-k+1)\)

Q: sampling formula for replacement:[...] ordered:[...]
=
[...]
A: NO; NO; \(\binom{n}{k}\)

Q: hockey-stick identity
[...] = [...]
A: \(\sum_{j=k}^n \binom{j}{k}\); \(\binom{n+1}{k+1}\)

Q: conditional probability
[...] = [...]
A: \(P(A|B)\); \(\frac{P(A \cap B)}{P(B)}\)

Q: random variable (r.v.)
A: function from sample space to real numbers

Q: probability mass function (PMF)
A: \(p_X(x) = P(X=x)\)

Q: true/false equal PMFs imply equal r.v.s
A: false

Q: Bernoulli dist: X ~ Bern(p)
A: P(X=1) = p
P(X=0) = 1 - p

Q: interpretation of Binomial distribution: Bin(n, p)
A: n independent Bern(p) trials

Q: Binomial distribution Bin(n,p) PMF
A: \[P(X=k)=\binom{n}{k} p^k (1-p)^{n-k}\]

Q: Hypergeometric dist. HGeom(w,b,n) interpretation
A: urn with w white balls, b black balls, n sampled with equal probability without replacement; number of white balls sampled

Q: Hypergeometric dist. HGeom(w,b,n) PMF
A: \[P(X=k)=\frac{\binom{w}{k} \binom{b}{n-k}}{\binom{w+b}{n}}\]

Q: how to think of hypergeometric distribution HGeom(w,b,n) in terms of "tags"
A: whether you're in w or b is one tag; whether you're sampled is another; thus the distribution HGeom(n, w+b-n, w) is equivalent

Q: discrete uniform dist. DUnif(C)
A: \[P(X=x)=\frac{1}{\vert C \vert}\]

Q: cumulative distribution function (CDF)
A: \[F_X(x)=P(X \leq x)\]

Q: conditions for when a function of an r.v. is an r.v.
A: always!

Q: r.v. independence criterion
A: \[P(X_1 \leq x_1, \ldots, X_n \leq x_n) = P(X_1 \leq x_1) \cdots P(X_n \leq x_n)\]for all \(x_1, \ldots, x_n \in \mathbb{R}\)

Q: relation of "independent" and "identically distributed"
A: you can have either without the other

Given:
- \(X \sim Bin(n, p)\)
- \(Y \sim Bin(m, p)\)
- X and Y are independent
Then:
- \(X+Y \sim Bin(n+m, p)\)
- conditional distribution of X given \(X+Y=r\) is \(HGeom(n,m,r)\)

Q: [...] = [...]
A: \(E(X+Y)\); \(E(X) + E(Y)\)

Q: conditions under which E(X+Y)=E(X)+E(Y)
A: always!

Q: Geom(p) distribution interpretation
A: Number of failures in p independent Bernoulli trials before the first success (note: in some texts, it's +1 to include the success)

Q: Geom(p) PMF
A: \[P(X=k)=q^kp\]

Q: negative binomial distribution NBin(r, p) interpretation
A: number of failures in independent Bern(p) trials before r successes

Q: NBin(r, p) PMF
A: \[P(X=n)={n + r - 1 \choose r - 1}p^rq^n\]

Q: indicator r.v.
A: 1 if a given event happened, 0 if not

Q: fundamental bridge between probability and expectation: [...] = [...]
A: \(P(A)\); \(E(I_A)\)

Q: Boole's or Bonferroni's inequality: [...] <= [...]
A: \(P(A_1 \cup \cdots \cup A_n)\); \(P(A_1) + \cdots + P(A_n)\)

Q: negative hypergeometric distribution NHGeom(w, b, r) interpretation
A: number of black balls drawn before drawing the rth white ball

Q: survival function
A: given CDF F, the survival function G(x) = 1 - F(x) = P(X > x)

Q: law of the unconscious statistician (LOTUS): [...] = [...]
A: \(E(g(X))\); \(\sum_x g(x)P(X=x)\)

Q: simpler formula for variance: Var(X) = [...]
A: \(E(X^2)-(EX)^2\)

Q: Var(X + c) = [...]
A: Var(X)

Q: Var(cX) = [...]
A: \(c^2 Var(X)\)

Q: conditions where Var(X + Y) = Var(X) + Var(Y)
A: X and Y are independent

Q: conditions where Var(X) = 0
A: iff P(X = a) = 1 for some constant a

Q: expected value of \(Pois(\lambda)\) distribution
A: \(\lambda\)

Q: variance of \(Pois(\lambda)\) distribution
A: \(\lambda\)

Q: Poisson distribution \(Pois(\lambda)\) PMF
A: \[P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}\]for k = 0, 1, 2, ...

Q: Poisson distribution \(Pois(\lambda)\) interpretation
A: "counting the number of successes in a particular region or interval of time, and there are a large number of trials, each with a small probability of success" .... lambda = "rate of occurrence" (Blitzstein/Hwang p. 177)

Q: distribution of the sum of two independent Poisson random variables
A: \[Pois(\lambda_1 + \lambda_2)\]

Q: PDF of Unif(a, b) distribution
A: \[\frac{1}{b-a}\]

Q: relation of CDF and PDF
A: CDF is integral of PDF (from negative infinity to x)
PDF is derivative of CDF

Q: definition of probability density function PDF
A: continuous r.v. has PDF f(x) if \(P(a \leq X \leq b) = \int_a^b f(x)dx\)
requires f(x) >= 0 and integral of f(x) from neg to pos infinity is 1

Q: [...] is the [...] probability of A
A: P(A); prior

Q: [...] is the [...] probability of A
A: P(A|B); posterior

Q: sensitivity vs specificity: in words
A: sensitivity = true positive rate
specificity = true negative rate

Q: sensitivity vs specificity: where T is even of a positive test and D is event of having the disease
A: sensitivity = \(P(T|D)\)
specificity = \(P(T^C|D^C)\)

Q: Bayes' rule with extra conditioning:
[...] = [...]
A: \(P(A|B,E)\); \(\frac{P(B|A,E)P(A|E)}{P(B|E)}\)

Q: true/false pairwise independence implies independence
A: false

Q: prosecutor's fallacy
A: confusing P(A|B) with P(B|A)

Q: defense attorney's fallacy
A: "failure to condition on all the evidence"

Q: Simpson's paradox explained in terms of doctors
A: having higher success rate in every category but lower overall success rate, because you're doing more of the harder category

Q: true/false: if X and Y are independent random variables, any function of X is independent of any function of Y
A: true

Q: [...] = [...]
A: E(cX); cE(X)

Q: how to get CDF F from PDF f
A: \[F(x) = \int_{-\infty}^{x} f(t)dt\]

Q: for a continuous r.v. PDF, [...] = [...]
A: \(P(a < X < b)\); \(P(a \leq X \leq b)\)

Q: for a continuous r.v. X with PDF f:
[...] = [...]
A: \(P(a < X \leq b)\); \(\int_a^b f(x)dx\)

Q: for continuous r.v. with PDF f:
[...] = [...]
A: E(X); \(\int_{-\infty}^{\infty}xf(x)dx\)

Q: Unif(a,b) PDF
A: \[f(x) = \begin{cases}\frac{1}{b-a} & \text{if } a < x < b, \\ 0 & \text{otherwise} \end{cases}\]

Q: if we condition on a Uniform r.v. being in a certain subinterval, what is the resulting distribution?
A: a Uniform distribution on that subinterval

Q: if X ~ Unif(a,b) and Y = cX + d, and [...], then [...]
A: c and d are constants with c > 0; Y ~ Unif(ca + d, cb + d)

Q: the [...] is called the [...]
A: inverse of the CDF; quantile function

Q: If U ~ Unif(0, 1) and X = [...], then [...].
A: \(F^{-1}(U)\); X is an r.v. with CDF F

Q: If X is [...], then F(X) ~ [...]
A: an r.v. with CDF F; Unif(0,1)

Q: relation of percentile and quantile
A: synonyms

Q: if X is an r.v. with CDF F, then [...] gives the [...]
A: F(X); percentile of X

Q: standard Normal distribution PDF aka \(\mathcal{N}(0,1)\)
A: \[\varphi(z) = \frac{1}{\sqrt{2\pi}}e^{-z^2/2}, -\infty<z<\infty\]

Q: true/false \(\varphi(z)=\varphi(-z)\)
A: true

Q: true/false if Z ~ \(\mathcal{N}(0,1)\) then -Z ~ \(\mathcal{N}(0,1)\)
A: true

Q: Normal distribution X ~ \(\mathcal{N}(\mu, \sigma^2)\) defined in terms of standard normal distribution
A: Z ~ \(N(0,1)\)
X = \(\mu + \sigma Z\)

Q: standardized version of X when X ~ \(\mathcal{N}(\mu, \sigma^2)\)
A: \[\frac{X - \mu}{\sigma} \sim \mathcal{N}(0,1)\]

Q: probability of being within one standard deviation of the mean in a Normal distribution
A: about 68%

Q: probability of being within two standard deviations of the mean in a Normal distribution
A: about 95%

Q: probability of being within three standard deviations of the mean in a Normal distribution
A: about 99.7%

Q: the [...] distribution is the continuous counterpart to the [...] distribution
A: Exponential; Geometric

Q: PDF for \(X \sim Expo(\lambda)\)
A: \[f(x)=\lambda e^{-\lambda x}, x >0\]

Q: true/false you can do location transformations on Exponential distribution
A: false

Q: true/false you can do scale transformations on Exponential distribution
A: true

Q: memoryless property
A: \[P(X \geq s + t | X \geq s) = P(X \geq t)\]for all s,t >= 0

Q: which continuous distributions on (0,infinity) are memoryless?
A: only the Exponential

Q: example of length-biased sampling
A: "asking randomly chosen mothers how many children they have yields a dilferent distribution from asking randomly chosen people how many siblings they have, including themselves"

Q: two conditions to have a Poisson process with rate \(\lambda\)
A: "1. The number of arrivals that occur in an interval of length t is a Pois(\(\lambda\)t) random variable.
2. The numbers of arrivals that occur in disjoint intervals are independent of each other...."

Q: [...] is called [...] of r.v. X
A: \(E(X^n)\); the nth moment

Q: median of a random variable X
A: c such that P(X <= c) >= 1/2 and P(X >= c) >= 1/2

Q: mode of a discrete r.v. X
A: c such that P(X = c) >= P(X = x) for all x

Q: mode of a continuous r.v. X
A: c such that f(x) >= f(x) for all x, where f ix the PDF

Q: the mean minimizes mean [...] error
A: squared

Q: the median minimizes mean [...] error
A: absolute

Q: formula for nth moment of r.v. X with mean \(\mu\) and variance \(\sigma^2\)
A: \[E(X^n)\]

Q: formula for nth central moment of r.v. X with mean \(\mu\) and variance \(\sigma^2\)
A: \[E((X-\mu)^n)\]

Q: formula for nth standardized moment of r.v. X with mean \(\mu\) and variance \(\sigma^2\)
A: \[E((\frac{X-\mu}{\sigma})^n)\]

Q: true/false for positive integer n, the nth moment of an r.v. must exist
A: false

Q: [...] is [...]
A: skewness; the third standardized moment

Q: an r.v. X is symmetric about \(\mu\) if...
A: \(X-\mu\) and \(\mu-X\) have the same distribution

Q: how to check for symmetry of continuous r.v. X with PDF f
A: symmetric about \(\mu\) iff \(f(x) = f(2\mu - x)\)

Q: if an r.v. is [...], every [...] [...]
A: symmetric; odd central moment; is 0 if it exists

Q: [...] is [...] but [...]
A: kurtosis; the fourth standardized moment; minus 3

Q: kth sample moment of X_1...X_n i.i.d. random variables
A: \[M_k = \frac{1}{n}\sum_j^nX_j^k\]

Q: If \(\bar{X}_n\)is the sample mean and \(\mu\) is the true mean, [...] = [...]
A: \(E(\bar{X}_n)\); \(\mu\)

Q: If \(\bar{X}_n\) is the sample mean, \(\mu\) is the true mean, and \(\sigma\) is the (true) standard deviation, then [...] = [...]
A: \(Var(\bar{X}_n)\); \(\frac{\sigma^2}{n}\)

Q: formula for sample variance of X_1...X_n i.i.d. random variables
A: \[S_n^2 = \frac{1}{n-1}\sum_j^n(X_j-\bar{X}_n)^2\]

Q: true/false sample variance is unbiased for estimating variance
A: true

Q: true/false sample standard deviation is unbiased for estimating standard deviation
A: false

Q: true/false every r.v. has a moment generating function
A: false

Q: for any valid moment generating function, [...] = [...]
A: M(0); 1

Q: general form of moment generating function of an r.v. X
A: \[M(t) = E(e^{tX})\]where t is "just a bookkeeping device"

Q: how to get nth moment from moment-generating function
A: evaluate the nth derivative of the MGF at 0

Q: how to tell two r.v.s have same distribution based on their moment-generating functions
A: "if there is even a tiny interval (-a, a) containing 0 on which the MGFs are equal"

Q: if X and Y are independent and M refers to moment-generating functions, [...] = [...]
A: \(M_{X+Y}(t)\); \(M_X(t)M_Y(t)\)

Q: alternative that always exists, unlike MGFs
A: characteristic function aka Fourier transform

Q: if M(t) is the MGF of X, what is the MGF of a + bX ?
A: \[e^{at}M(bt)\]

Q: Log-Normal distribution (in words)
A: the log of the r.v. follows Normal distribution
(it's NOT "log of a Normal")

Q: Log-Normal distribution: Y ~ \(\mathcal{L}\mathcal{N}(\mu, \sigma^2)\)
A: X ~ \(\mathcal{N}(\mu,\sigma^2)\)
Y = \(e^X\)

Q: Weibull distribution, given \(\lambda,\gamma > 0\)
A: X ~ Expo(\(\lambda\))
T = \(X^{1/\gamma}\)

Q: marginal vs conditional distribution
A: marginal = unconditional distribution of one variable, ignoring other variables
conditional = distribution of one variable, conditioned on values of other variables

Q: marginal PMF of X for discrete r.v.s X and Y
A: \[P(X=x)=\sum_y P(X=x,Y=y)\]

Q: conditional PMF of Y given X=x
A: \[P(Y=y|X=x)=\frac{P(X=x,Y=y)}{P(X=x)}\]

Q: does the joint PMF equal the product of the marginal PMFs?
A: iff the variables are independent

Q: independence implies that [...] are equal to [...]
A: all the conditional PMFs; the marginal PMF

Q: If the joint CDF for two continuous r.v.s is:
[...]
then the joint PDF is:
[...]
A: \(F_{X,Y}(x,y)=P(X\leq x, Y \leq y)\); \(f_{X,Y}(x,y)=\frac{\partial^2}{\partial_x\partial_y}F_{X,Y}(x,y)\)

Q: marginal PDF of X, where X and Y are continuous r.v.s with joint PDF \(f_{X,Y}\)
A: \[f_X(x)=\int_{-\infty}^\infty f_{X,Y}(x,y)dy\]

Q: conditional PDF of Y given X = x, where X and Y are continuous r.v.s with joint PDF \(f_{X,Y}\)
A: \[f_{Y|X}(y|x)=\frac{f_{X,Y}(x,y)}{f_X(x)}\]

If a joint PDF factors into g(x)h(x) then we know X and Y are independent, as long as g and h are nonnegative functions and the factoring holds for all x and y.
Further, if g or h is a valid PDF, then both g and h are valid PDFs and they are the marginal PDFs of X and Y.

Q: Cauchy distribution
A: X/Y where X and Y are drawn i.i.d from standard normal

Q: two formulas for covariance
A: \[Cov(X,Y)=E((X-EX)(Y-EY))\]\[Cov(X,Y)=E(XY)-E(X)E(Y)\]

Q: does independent imply uncorrelated?
A: yes

Q: does uncorrelated imply independent?
A: no
"r.v.s can be dependent in nonlinear ways and still have zero covariance", as with X~N(0,1) and Y=X^2

Q: Cov(X,[...])=[...]
A: X; Var(X)

Q: is Cov(X,Y) = Cov(Y,X)?
A: yes

Q: Cov(X,c) for any constant c
A: 0

Q: [...] = [...] for any constant a
A: Cov(aX, Y; aCov(X,Y

Q: [...] = [...]
A: Cov(X+Y, Z); Cov(X,Z) + Cov(Y,Z)

Q: [...] = [...]
A: Cov(X+Y, Z+W); Cov(X,Z) + Cov(X,W) + Cov(Y,Z) + Cov(Y,W)

relating variance and covariance:
Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)
or more generally:
\(Var(X_1+\cdots+X_n)\) = \(Var(X_1)+\cdots+Var(X_n)+2\sum_{i<j}Cov(X_i,X_j)\)

Q: formula for correlation
A: \[Corr(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}\]

Q: advantage of correlation compared to covariance
A: "convenient to interpret because it does not depend on the units of measurement and is always between -1 and 1"

Q: Corr(cX, Y) = [...]
A: Corr(X,Y)

Q: description of Multinomial distribution
A: parameters: k (number of categories), n (number of objects, which will each be placed independently into one of the categories), vector p (probability for each category, summing to 1)
joint PMF gives probability for a vector specifying the number of objects in each category

Q: Multinomial joint PMF: \(\mathbf{X} \sim Mult_k(n,\mathbf{p})\)
A: \[\frac{n!}{n_1!n_2!\ldots n_k!}p_1^{n_1}p_2^{n^2}\cdots p_k^{n_k}\]for \(n_1,\ldots,n_k\) satisfying \(n_1+\cdots+n_k = n\)

Q: marginal of a Multinomial:
if \(\mathbf{X} \sim Mult_k(n, \mathbf{p})\) then \(X_j \sim\) [...]
A: \(Bin(n,p_j)\)

Q: Multinomial lumping:
If \(\mathbf{X}\sim Mult_k(n,\mathbf{p})\), then for any distinct i and j:
\(X_i + X_j \sim \) [...]
and new multinomial with categories \(X_i, X_j\) merged can be formed by [...]
A: \(Bin(n, p_i+p_j)\)); using k-1 and p1 + p2

Q: Multinomial conditioning:
If \(\mathbf{X} \sim Mult_k(n, \mathbf{p})\), then:
\((X_2,\ldots,X_k)|X_1 =\) [...]
A: \(n_1 \sim Mult_{k-1}(n-n_1,(p'_2,\ldots,p'_k))\) where \(p'_j=p_j/(p_2+\cdots+p_k)\)

Q: Covariance in a Multinomial:
If \(\mathbf{X} \sim Mult_k(n,\mathbf{p})\), then for \(i \neq j\):
\(Cov(X_i, X_j) =\) [...]
A: \(-np_ip_j\)

Q: a k-dimensional random vector has Multivariate Normal distribution if...
A: every linear combination of its elements has a Normal distribution

Q: does each element of an MVN vector have marginal distribution Normal?
A: yes

Q: if all elements of a vector have Normal distribution, is its joint distribution necessarily MVN?
A: no
example: X is normal, S has 50/50 chance of being 1 or -1; SX is normal but (X, SX) is not MVN

Q: is every subvector of an MVN also an MVN?
A: yes

Q: is the concatenation of two MVNs an MVN?
A: yes

Q: to fully specify an MVN you need both [...] and [...]
A: the mean vector; the covariance matrix

Q: if two components of an MVN are uncorrelated, are they necessarily independent?
A: yes

Q: Change of Variables formula and its assumptions
(in 1 dimension)
X is a continuous r.v. with PDF \(f_X\), and \(Y = g(X)\)
A: Assuming: g is differentiable, and g is strictly increasing or strictly decreasing
Then: the PDF of Y is \(f_Y(y) = f_X(x)|\frac{dx}{dy}|\), where \(x=g^{-1}(y)\)

Q: PDF of a location-scale transformation
where X has PDF \(f_X\) and \(Y = a+bX\)
A: \(f_Y(y) = f_X(\frac{y-a}{b})\frac{1}{|b|}\)
where \(y = a+bx\)

Q: (probability): a convolution is...
A: a sum of independent random variables

Q: if X and Y are independent discrete r.v.s and T = X + Y, what is the PMF of T?
A: \(P(T=t)=\sum_yP(X=t-y)P(Y=y)\)

Q: if X and Y are independent continuous r.v.s and T = X + Y, what is the PDF of T?
A: \(f_T(t) = \int_{-\infty}^{\infty}f_X(t-y)f_Y(y)dy\)

Q: connection between Beta and Unif distributions
A: - Beta is a generalization of Unif(0,1)
- Beta(1,1) is same as Unif(0,1)

Q: relation between Beta and Binomial distributions
A: "the Beta is the conjugate prior of the Binomial"
"if we have a Beta prior distribution on p and data that are conditionally Binomial given p, then when going from prior to posterior, we don't leave the family of Beta distributions"

Q: [...] = [...]
A: (gamma function) \(\Gamma(a)\); \(\int_0^\infty x^ae^{-x}\frac{dx}{x}\)

Q: [...] = [...]
(a Gamma function property)
A: \(\Gamma(a+1)\); \(a\Gamma(a)\)

Q: [...] = [...] if [...]
(a gamma function property
A: \(\Gamma(n)\); \((n-1)!\); n is a positive integer

Q: relation of Gamma and Exponential distributions
A: - Gamma is a generalization of Expo
- \(Gamma(1,\lambda)\) is same as \(Expo(\lambda)\)

Q: If \(X_1,\ldots,X_n\) are i.i.d. Expo(\(\lambda\)), then:
[...] ~ [...]
A: \(X_1+\cdots+X_n\); Gamma(n, \(\lambda\))

Q: [...] represents [...] in a Poisson process of rate \(\lambda\)
A: Y ~ Gamma(a, \(\lambda\)); the total waiting time for the ath arrival

Q: "the [...] is the continuous analog of the [...]"
A: Gamma distribution; Negative Binomial distribution

Q: conjugate prior of the Poisson distribution
A: Gamma distribution

Q: what are order statistics?
A: the r.v.s \(min(X_1,\ldots,X_n),\ldots,max(X_1,\ldots,X_n)\). denoted \(X_{(1)},\ldots,X_{(n)}\)

Q: CDF of order statistic \(X_{(j)}\) if \(X_1,\ldots,X_n\) are i.i.d. continuous r.v.s with CDF F
A: \(P(X_{(j)}\leq x) = \sum_{k=j}^n\binom{n}{k}F(x)^k(1-F(x))^{n-k}\)

Q: conditional expectation given an event vs conditional expectation given a random variable
A: when X is an r.v., E(Y|X) is itself an r.v.

Q: why is the expected waiting time for HH in a series of fair coin tosses higher than the expected waiting time for HT?
A: with HT, you have to start over when you see a T

Q: if X and Y are independent, then E(Y|X) = ...
A: E(Y)

Q: for any function h, [...] = [...]
A: \(E(h(X)Y|X)\); \(h(X)E(Y|X)\)

Q: [...] = [...]
A: \(E(Y_1+Y_2|X)\); \(E(Y_1|X)+E(Y_2|X)\)

Q: E(E(Y|X)) = [...]
A: E(Y)

Q: "The r.v. [...], which is called the residual from using X to predict Y, is [...] for any function h."
A: Y - E(Y|X); uncorrelated with h(x)

Q: conditional variance of Y given X
A: \(Var(Y|X) = E(Y^2|X) - (E(Y|X))^2\)

Q: Eve's law
A: for any r.v.s X and Y,
\(Var(Y) = E(Var(Y|X)) + Var(E(Y|X))\)

Q: how two kinds of variation are accounted for in Eve's law
A: within-group (or unexplained) variation = E(Var(Y|X))
between-group (or explained) variation = Var(E(Y|X))

Q: Cauchy-Schwarz inequality in probability:
[...] \(\leq\) [...]
A: \(|E(XY)|\); \(\sqrt{E(X^2)E(Y^2)}\)

Q: Jensen's inequality for concave functions
A: E(g(X)) <= g(E(X))

Q: Jensen's inequality for convex functions
A: E(g(X)) >= g(E(X))

Q: Jensen's inequality can be used to show that sample standard deviation [...] the true standard deviation
A: underestimates

Q: Markov's inequality: for any r.v. X and constant a > 0...
A: \[P(|X|\geq a) \leq \frac{E|X|}{a}\]

Q: Markov's inequality interpretation
A: you can't have more than half of people making at least twice the average income; or more than 1/3 of people making at least three times the average income; etc

Q: Chebysev's inequality: X has mean \(\mu\) and variance \(\sigma^2\); then for any a > 0...
A: \[P(|X-\mu|\geq a)\leq \frac{\sigma^2}{a^2}\]

Q: Chebysev's inequality interpretation when formulated as [...]: [...]
A: \(P(|X-\mu|\geq cq)\leq\frac{1}{c^2}\); upper bound on probability of X being more than c standard deviations away from the mean

Q: Chernoff bound: for any r.v. X and constants a > 0 and t > 0...
A: \[P(X\geq a) \leq \frac{E(e^{tX})}{e^{ta}}\]

Q: law of large numbers
A: the sample mean converges to the true mean as the sample size grows

Q: summary of Monte Carlo integration
A: You want to approximate an integral \(\int_a^bf(x)dx\) where you know \(0 \leq f(x) \leq c\). Imagine a rectangle with the x-axis running from a to b and the y-axis running from 0 to c. You sample points from this rectangle randomly and see what proportion fall beneath the curve.

Q: empirical CDF \(\hat{F}_n\) for n i.i.d. random variables
A: given \(R_n(x) = \sum_{j=1}^nI(X_j\leq x)\),
\(\hat{F}_n(x) = \frac{R_n(x)}{n}\)

Q: central limit theorem, given \(X_1,X_2,X_3,\ldots\) drawn i.i.d. with mean \(\mu\) and variance \(\sigma^2\)
A: as \(n \rightarrow \infty\):
\(\sqrt{n}(\frac{\bar{X}_n-\mu}{\sigma}) \rightarrow \mathcal{N}(0,1)\) in distribution
(i.e. the after-standardization distribution of the sample mean approaches a standard Normal distribution)

Q: central limit theorem, approximation form
A: for large n, the distribution of \(\bar{X}_n\) is approximately \(\mathcal{N}(\mu, \frac{\sigma^2}{n})\)
(i.e. the sample mean approximately follows a normal distribution based on mean, variance, and sample size)

Q: preconditions for law of large numbers and central limit theorem that aren't always met
A: mean and variance must be finite

Q: a distribution where central limit theorem and law of large numbers don't apply
A: Cauchy, because it has no mean or variance
(sample mean of n Cauchys is always drawn from Cauchy)

Q: how central limit theorem can be used to create Normal distribution-based approximations of other distributions
A: when the distribution can be represented as a sum of i.i.d. r.v.s, you can use the fact that \(X_1+\ldots+X_n=n\bar{X}_n\)

Q: relation of Chi-Square and Gamma distributions
A: \(\chi_n^2\) is same as Gamma(\(\frac{n}{2},\frac{1}{2}\))

Q: the Chi-Square distribution with n degrees of freedom is written [...]
V = [...]
A: \(V \sim \chi_n^2\); \(Z_1^2+\ldots+Z_n^2\) where \(Z_1,Z_2,\ldots,Z_n\) are i.i.d. \(\mathcal{N}(0,1)\)

Q: Student-t distribution with n degrees of freedom: T ~ \(t_n\)
A: given \(Z \sim \mathcal{N}(0,1)\), \(V \sim \chi_n^2\), and Z independent of V
\(T = \frac{Z}{\sqrt{V/n}}\)

Q: three properties of Student-t distribution
A: 1. Symmetry: if T ~ \(t_n\) then -T ~ \(t_n\)
2. \(t_1\) is the Cauchy distribution
3. as \(n \rightarrow \infty\), \(t_n\) approaches the standard Normal distribution

Q: what does 'recurrent' mean in the context of Markov chains?
A: that if you start from a given state, you will (with probability 1) eventually return to that state

Q: what does 'transient' mean in the context of Markov chains?
A: not recurrent
i.e., if you start from a given state, the probability of eventually returning to it is less than 1

Q: the number of returns to a transient state in a Markov chain follows what distribution?
A: Geometric

Q: what do 'reducible' and 'irreducible' mean in the context of Markov chains?
A: a chain is irreducible if, for any two states, it's possible to go between them in a finite number of steps (with probability > 0)
otherwise it's reducible

Q: what's the relation between reducibility and recurrence, in the context of Markov chains?
A: - recurrence is a property of states, reducibility is a property of chains
- in an irreducible chain, all states are recurrent; but it's possible for all states be recurrent in a reducible chain too, if there are separate 'islands' that can't reach each other

Q: how do you find the probability of going from state i to state j in n steps in a Markov chain?
A: take the nth power of the transition matrix then look at cell i,j

Q: relation of periodic state to periodic chain, in the context of Markov chains
A: a chain is aperiodic if all its states are aperiodic, otherwise it's periodic

Q: define "period" of a state of a Markov chain
A: greatest common divisor of the numbers of steps it can take to return to the state when starting from that state

Q: periodic vs aperiodic state of a Markov chain
A: aperiodic means its period = 1

Q: what does the fact that a Markov chain is irreducible tell you about the periods of its states?
A: they all have the same period

Q: stationary distribution of a Markov chain with transition matrix Q
A: row vector s such that \(s_i \geq 0\), \(\sum_i s_i = 1\), and \(\sum_i s_i q_{ij} = s_j\) for all j
can also be written sQ=s

Q: for Markov chain in a finite state space, is there guaranteed to be a stationary distribution?
A: yes

Q: the stationary distribution of a Markov chain is guaranteed to be unique when...
A: the chain is irreducible

Q: for a Markov chain with transition matrix Q, as \(n \rightarrow \infty\), [...], if [...]
A: \(Q^n\) converges to a matrix where each row is the stationary distribution; the chain is irreducible and aperiodic

Q: for an irreducible Markov chain with stationary distribution s, if \(r_i\) is [...], then \(s_i\) = [...]
A: the expected time it takes the chain to return to i given that it starts at i; \(\frac{1}{r_i}\)

Q: a Markov chain with transition matrix Q is reversible if...
A: there's a vector s with \(s_i \geq 0\), \(\sum_i s_i = 1\), such that \(s_iq_{ij} = s_jq_{ji}\)

Q: true/false all reversible Markov chains have a stationary distribution
A: true

Q: if each column of a Markov chain's column [...], then [...]
A: sums to 1; (1/M, ..., 1/M) is a stationary distribution

Q: Metropolis-Hastings algorithm, general idea
A: Turns any irreducible Markov chain into a chain with a specified stationary distribution. This is done by deciding, at each step, whether to actually transition or not by using an 'acceptance probability' calculated from the stationary distribution and original transition matrix.
This is useful because you can then use the chain to sample from the distribution, which might have been hard to do directly.

Q: Metropolis-Hastings acceptance probability formula, given stationary distribution s and original transition matrix P
A: \[a_{ij} = \text{min}(\frac{s_jp_{ji}}{s_ip_{ij}}, 1)\]

Q: describe Gibbs sampling
A: a Markov-chain Monte Carlo algorithm for (approximately) sampling from a joint distribution
it updates one component at a time holding the others constant

Q: how Poisson, Expo, and Gamma distributions are relevant to a Poisson process
A: - number of arrivals in an interval of length t is distributed Pois(\(\lambda t\)) (note: numbers of arrivals in disjoint time intervals must be independent)
- interarrival times: \(T_j-T_{j-1} \sim \text{Expo}(\lambda)\)
- arrival times: \(T_j\sim \text{Gamma}(j,\lambda)\)

Q: "In a Poisson process of rate \(\lambda\), conditional on [...], the joint distribution of the arrival times \(T_1,\ldots,T_n\) is the same as [...]"
A: \(N(t) = n\); the joint distribution of the order statistics of n i.i.d. \(\text{Unif}(0,t)\) r.v.s.

Q: Given two independent Poisson processes \((N_1(t) : t > 0)\) and \((N_2(t) : t > 0)\) with rates \(\lambda_1\) and \(\lambda_2\), if you overlay them you get...
A: a combined Poisson process \(N(t) = N_1(t) + N_2(t)\) with rate \(\lambda_1 + \lambda_2\)

Q: competing risks theorem summary
A: given \(Y_1 \sim \text{Expo}(\lambda_1)\) and \(Y_2 \sim \text{Expo}(\lambda_2)\):
\(\text{min}(Y_1,Y_2)\) is independent of \(I(Y_1 < Y_2)\)
so e.g. if you have two appliances with different average lifetimes and you know when one first failed, this gives you no information about which one failed

Q: explain the "thinning" property of Poisson processes
A: given one Poisson process with rate \(\lambda\), you can split it into two independent Poisson processes of rates \(\lambda p\) and \(\lambda(1-p)\) by independently flipping a coin of probability p at each arrival