{
  "deckName": "Exam P — Formula Sheet",
  "examCode": "Exam P",
  "cards": [
    {
      "front": "Probability axioms (Kolmogorov) — three rules.",
      "back": "$\\Pr(A)\\geq 0$ $\\cdot$ $\\Pr(S)=1$ $\\cdot$ countable additivity: disjoint $A_i\\Rightarrow\\Pr\\!\\left(\\bigcup_i A_i\\right)=\\sum_i\\Pr(A_i)$.",
      "tag": "Core probability"
    },
    {
      "front": "Complement and addition (inclusion–exclusion, 2 events).",
      "back": "$\\Pr(A^{c})=1-\\Pr(A)$ $\\cdot$ $\\Pr(A\\cup B)=\\Pr(A)+\\Pr(B)-\\Pr(A\\cap B)$.",
      "tag": "Core probability"
    },
    {
      "front": "Inclusion–exclusion for 3 events.",
      "back": "$\\Pr(A\\cup B\\cup C)=\\Pr(A)+\\Pr(B)+\\Pr(C)-\\Pr(A\\cap B)-\\Pr(A\\cap C)-\\Pr(B\\cap C)+\\Pr(A\\cap B\\cap C)$.",
      "tag": "Core probability"
    },
    {
      "front": "Conditional probability, multiplication rule, independence.",
      "back": "$\\Pr(A\\mid B)=\\dfrac{\\Pr(A\\cap B)}{\\Pr(B)}$ $\\cdot$ $\\Pr(A\\cap B)=\\Pr(A\\mid B)\\Pr(B)$ $\\cdot$ independent $\\Leftrightarrow\\Pr(A\\cap B)=\\Pr(A)\\Pr(B)$.",
      "tag": "Core probability"
    },
    {
      "front": "Law of total probability and Bayes' theorem (partition $\\{B_i\\}$).",
      "back": "$\\Pr(A)=\\sum_i\\Pr(A\\mid B_i)\\Pr(B_i)$ $\\cdot$ $\\Pr(B_k\\mid A)=\\dfrac{\\Pr(A\\mid B_k)\\Pr(B_k)}{\\sum_i\\Pr(A\\mid B_i)\\Pr(B_i)}$.",
      "tag": "Core probability"
    },
    {
      "front": "Bernoulli$(p)$: pmf, support, mean, variance, MGF.",
      "back": "$p(x)=p^{x}(1-p)^{1-x}$, $x\\in\\{0,1\\}$ $\\cdot$ $E[X]=p$ $\\cdot$ $\\operatorname{Var}(X)=p(1-p)$ $\\cdot$ $M_X(t)=q+pe^{t}$, $q=1-p$.",
      "tag": "Discrete dist"
    },
    {
      "front": "Binomial$(n,p)$: pmf, support, mean, variance, MGF.",
      "back": "$p(x)=\\binom{n}{x}p^{x}q^{\\,n-x}$, $x=0,\\dots,n$ $\\cdot$ $E[X]=np$ $\\cdot$ $\\operatorname{Var}(X)=npq$ $\\cdot$ $M_X(t)=(q+pe^{t})^{n}$, $q=1-p$.",
      "tag": "Discrete dist"
    },
    {
      "front": "Geometric$(p)$, trials-to-first-success form: pmf, support, mean, variance, MGF.",
      "back": "$p(x)=q^{x-1}p$, $x=1,2,\\dots$ $\\cdot$ $E[X]=\\dfrac{1}{p}$ $\\cdot$ $\\operatorname{Var}(X)=\\dfrac{q}{p^{2}}$ $\\cdot$ $M_X(t)=\\dfrac{pe^{t}}{1-qe^{t}}$, $q=1-p$.",
      "tag": "Discrete dist"
    },
    {
      "front": "Geometric — failures-before-first-success form ($Y=X-1$): pmf, support, mean, variance.",
      "back": "$p(y)=q^{y}p$, $y=0,1,2,\\dots$ $\\cdot$ $E[Y]=\\dfrac{q}{p}$ $\\cdot$ $\\operatorname{Var}(Y)=\\dfrac{q}{p^{2}}$ (same variance; mean drops by $1$).",
      "tag": "Discrete dist"
    },
    {
      "front": "Negative Binomial$(r,p)$, trials-to-$r$th-success form: pmf, support, mean, variance, MGF.",
      "back": "$p(x)=\\binom{x-1}{r-1}p^{r}q^{\\,x-r}$, $x=r,r+1,\\dots$ $\\cdot$ $E[X]=\\dfrac{r}{p}$ $\\cdot$ $\\operatorname{Var}(X)=\\dfrac{rq}{p^{2}}$ $\\cdot$ $M_X(t)=\\left(\\dfrac{pe^{t}}{1-qe^{t}}\\right)^{r}$.",
      "tag": "Discrete dist"
    },
    {
      "front": "Negative Binomial — failures-before-$r$th-success form ($W=X-r$): pmf, support, mean, variance.",
      "back": "$p(w)=\\binom{w+r-1}{w}p^{r}q^{w}$, $w=0,1,\\dots$ $\\cdot$ $E[W]=\\dfrac{rq}{p}$ $\\cdot$ $\\operatorname{Var}(W)=\\dfrac{rq}{p^{2}}$.",
      "tag": "Discrete dist"
    },
    {
      "front": "Hypergeometric$(N,K,n)$: pmf, support, mean, variance.",
      "back": "$p(x)=\\dfrac{\\binom{K}{x}\\binom{N-K}{\\,n-x}}{\\binom{N}{n}}$ $\\cdot$ $E[X]=n\\dfrac{K}{N}$ $\\cdot$ $\\operatorname{Var}(X)=n\\dfrac{K}{N}\\!\\left(1-\\dfrac{K}{N}\\right)\\!\\dfrac{N-n}{N-1}$ (last factor = finite-pop correction).",
      "tag": "Discrete dist"
    },
    {
      "front": "Poisson$(\\lambda)$: pmf, support, mean, variance, MGF.",
      "back": "$p(x)=\\dfrac{e^{-\\lambda}\\lambda^{x}}{x!}$, $x=0,1,2,\\dots$ $\\cdot$ $E[X]=\\lambda$ $\\cdot$ $\\operatorname{Var}(X)=\\lambda$ $\\cdot$ $M_X(t)=\\exp[\\lambda(e^{t}-1)]$.",
      "tag": "Discrete dist"
    },
    {
      "front": "Poisson facts: factorial moment, Poisson limit of binomial, thinning.",
      "back": "$E[X(X-1)]=\\lambda^{2}$ $\\cdot$ $\\text{Bin}(n,p)\\to\\text{Poisson}(np)$ as $n\\to\\infty,\\,p\\to0$ $\\cdot$ split with prob $p$: independent $\\text{Poisson}(\\lambda p)$ and $\\text{Poisson}(\\lambda(1-p))$.",
      "tag": "Discrete dist"
    },
    {
      "front": "Discrete Uniform on $\\{1,\\dots,n\\}$: pmf, mean, variance.",
      "back": "$p(x)=\\dfrac{1}{n}$ $\\cdot$ $E[X]=\\dfrac{n+1}{2}$ $\\cdot$ $\\operatorname{Var}(X)=\\dfrac{n^{2}-1}{12}$.",
      "tag": "Discrete dist"
    },
    {
      "front": "Continuous Uniform$(a,b)$: pdf, cdf, mean, variance, MGF.",
      "back": "$f(x)=\\dfrac{1}{b-a}$ on $[a,b]$ $\\cdot$ $F(x)=\\dfrac{x-a}{b-a}$ $\\cdot$ $E[X]=\\dfrac{a+b}{2}$ $\\cdot$ $\\operatorname{Var}(X)=\\dfrac{(b-a)^{2}}{12}$ $\\cdot$ $M_X(t)=\\dfrac{e^{tb}-e^{ta}}{t(b-a)}$.",
      "tag": "Continuous dist"
    },
    {
      "front": "Exponential (rate $\\lambda$, mean $\\theta=1/\\lambda$): pdf, cdf, survival, mean, variance, MGF.",
      "back": "$f(x)=\\lambda e^{-\\lambda x}$, $x\\geq0$ $\\cdot$ $F(x)=1-e^{-\\lambda x}$ $\\cdot$ $S(x)=e^{-\\lambda x}$ $\\cdot$ $E[X]=\\dfrac{1}{\\lambda}=\\theta$ $\\cdot$ $\\operatorname{Var}(X)=\\dfrac{1}{\\lambda^{2}}=\\theta^{2}$ $\\cdot$ $M_X(t)=\\dfrac{\\lambda}{\\lambda-t}$, $t<\\lambda$.",
      "tag": "Continuous dist"
    },
    {
      "front": "Exponential memorylessness and second moment.",
      "back": "$\\Pr(X>s+t\\mid X>s)=\\Pr(X>t)$ $\\cdot$ $E[X^{2}]=2\\theta^{2}$ $\\cdot$ only continuous memoryless distribution.",
      "tag": "Continuous dist"
    },
    {
      "front": "Gamma (shape $\\alpha$, rate $\\lambda$): pdf, mean, variance, MGF.",
      "back": "$f(x)=\\dfrac{\\lambda^{\\alpha}}{\\Gamma(\\alpha)}x^{\\alpha-1}e^{-\\lambda x}$, $x>0$ $\\cdot$ $E[X]=\\dfrac{\\alpha}{\\lambda}$ $\\cdot$ $\\operatorname{Var}(X)=\\dfrac{\\alpha}{\\lambda^{2}}$ $\\cdot$ $M_X(t)=\\left(\\dfrac{\\lambda}{\\lambda-t}\\right)^{\\alpha}$, $t<\\lambda$.",
      "tag": "Continuous dist"
    },
    {
      "front": "Gamma facts: gamma function, Erlang, exponential and chi-square links.",
      "back": "$\\Gamma(\\alpha+1)=\\alpha\\Gamma(\\alpha)$, $\\Gamma(n)=(n-1)!$, $\\Gamma\\!\\left(\\tfrac12\\right)=\\sqrt{\\pi}$ $\\cdot$ $\\alpha=1$: exponential $\\cdot$ integer $\\alpha=n$: sum of $n$ iid $\\text{Exp}(\\lambda)$ (Erlang) $\\cdot$ $\\chi^{2}_{k}=\\text{Gamma}(\\alpha=k/2,\\lambda=\\tfrac12)$.",
      "tag": "Continuous dist"
    },
    {
      "front": "Normal$(\\mu,\\sigma^{2})$: pdf, mean, variance, MGF, standardization.",
      "back": "$f(x)=\\dfrac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\!\\left(-\\dfrac{(x-\\mu)^{2}}{2\\sigma^{2}}\\right)$ $\\cdot$ $E[X]=\\mu$, $\\operatorname{Var}(X)=\\sigma^{2}$ $\\cdot$ $M_X(t)=\\exp\\!\\left(\\mu t+\\tfrac12\\sigma^{2}t^{2}\\right)$ $\\cdot$ $Z=\\dfrac{X-\\mu}{\\sigma}\\sim N(0,1)$.",
      "tag": "Continuous dist"
    },
    {
      "front": "Beta$(\\alpha,\\beta)$ on $[0,1]$: pdf, mean, variance.",
      "back": "$f(x)=\\dfrac{\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}x^{\\alpha-1}(1-x)^{\\beta-1}$, $0<x<1$ $\\cdot$ $E[X]=\\dfrac{\\alpha}{\\alpha+\\beta}$ $\\cdot$ $\\operatorname{Var}(X)=\\dfrac{\\alpha\\beta}{(\\alpha+\\beta)^{2}(\\alpha+\\beta+1)}$ $\\cdot$ Beta$(1,1)=U(0,1)$.",
      "tag": "Continuous dist"
    },
    {
      "front": "Beta function and its gamma identity.",
      "back": "$B(\\alpha,\\beta)=\\int_{0}^{1}x^{\\alpha-1}(1-x)^{\\beta-1}\\,dx=\\dfrac{\\Gamma(\\alpha)\\Gamma(\\beta)}{\\Gamma(\\alpha+\\beta)}$.",
      "tag": "Continuous dist"
    },
    {
      "front": "Survival, hazard rate, and tail-integral mean (continuous).",
      "back": "$S(x)=1-F(x)$ $\\cdot$ $h(x)=\\dfrac{f(x)}{S(x)}=-\\dfrac{d}{dx}\\ln S(x)$, so $S(x)=\\exp\\!\\left(-\\int_{0}^{x}h(t)\\,dt\\right)$ $\\cdot$ $E[X]=\\int_{0}^{\\infty}S(x)\\,dx$ ($X\\geq0$).",
      "tag": "Expectation"
    },
    {
      "front": "Expectation: definition, LOTUS, linearity.",
      "back": "$E[X]=\\int x f(x)\\,dx$ or $\\sum x\\,p(x)$ $\\cdot$ LOTUS: $E[g(X)]=\\int g(x)f(x)\\,dx$ $\\cdot$ $E[aX+b]=aE[X]+b$ $\\cdot$ $E[X+Y]=E[X]+E[Y]$ (always).",
      "tag": "Expectation"
    },
    {
      "front": "Variance: shortcut, linear transform, and Jensen.",
      "back": "$\\operatorname{Var}(X)=E[X^{2}]-(E[X])^{2}$ $\\cdot$ $\\operatorname{Var}(aX+b)=a^{2}\\operatorname{Var}(X)$, $\\operatorname{SD}=|a|\\sigma$ $\\cdot$ convex $g$: $E[g(X)]\\geq g(E[X])$ (e.g. $E[X^{2}]\\geq(E[X])^{2}$).",
      "tag": "Expectation"
    },
    {
      "front": "Coefficient of variation, skewness, kurtosis.",
      "back": "$\\operatorname{CV}=\\dfrac{\\sigma}{\\mu}$ ($=1$ for exponential) $\\cdot$ skewness $\\gamma_1=\\dfrac{E[(X-\\mu)^{3}]}{\\sigma^{3}}$ $\\cdot$ kurtosis $\\dfrac{E[(X-\\mu)^{4}]}{\\sigma^{4}}$ (normal $=3$; excess subtracts $3$).",
      "tag": "Expectation"
    },
    {
      "front": "MGF: definition, moment recovery, value at 0, uniqueness, $aX+b$ rule.",
      "back": "$M_X(t)=E[e^{tX}]$ $\\cdot$ $E[X^{n}]=M_X^{(n)}(0)$ $\\cdot$ $M_X(0)=1$ $\\cdot$ MGF (on an interval around $0$) determines the distribution $\\cdot$ $M_{aX+b}(t)=e^{bt}M_X(at)$.",
      "tag": "MGF/PGF"
    },
    {
      "front": "MGF of independent sums; PGF basics.",
      "back": "Independent: $M_{\\sum X_i}(t)=\\prod_i M_{X_i}(t)$ $\\cdot$ PGF $G_X(s)=E[s^{X}]$ (integer $X\\geq0$) $\\cdot$ $G_X'(1)=E[X]$, $G_X''(1)=E[X(X-1)]$, $\\Pr(X=k)=\\dfrac{G_X^{(k)}(0)}{k!}$.",
      "tag": "MGF/PGF"
    },
    {
      "front": "Covariance: definition, shortcut, with itself, under independence.",
      "back": "$\\operatorname{Cov}(X,Y)=E[(X-\\mu_X)(Y-\\mu_Y)]=E[XY]-E[X]E[Y]$ $\\cdot$ $\\operatorname{Cov}(X,X)=\\operatorname{Var}(X)$ $\\cdot$ independent $\\Rightarrow\\operatorname{Cov}=0$ (converse false).",
      "tag": "Covariance/correlation"
    },
    {
      "front": "Covariance bilinearity and correlation.",
      "back": "$\\operatorname{Cov}(aX+b,\\,cY+d)=ac\\,\\operatorname{Cov}(X,Y)$ $\\cdot$ $\\operatorname{Cov}\\!\\left(\\sum_i a_iX_i,\\sum_j b_jY_j\\right)=\\sum_{i,j}a_ib_j\\operatorname{Cov}(X_i,Y_j)$ $\\cdot$ $\\rho=\\dfrac{\\operatorname{Cov}(X,Y)}{\\sigma_X\\sigma_Y}\\in[-1,1]$.",
      "tag": "Covariance/correlation"
    },
    {
      "front": "Variance of a sum / difference / linear combination.",
      "back": "$\\operatorname{Var}(X\\pm Y)=\\operatorname{Var}(X)+\\operatorname{Var}(Y)\\pm2\\operatorname{Cov}(X,Y)$ $\\cdot$ $\\operatorname{Var}\\!\\left(\\sum_i a_iX_i\\right)=\\sum_i a_i^{2}\\operatorname{Var}(X_i)+2\\sum_{i<j}a_ia_j\\operatorname{Cov}(X_i,X_j)$.",
      "tag": "Covariance/correlation"
    },
    {
      "front": "Joint, marginal, conditional, and independence (continuous).",
      "back": "$f_X(x)=\\int f(x,y)\\,dy$ $\\cdot$ $f_{Y\\mid X}(y\\mid x)=\\dfrac{f(x,y)}{f_X(x)}$ $\\cdot$ independent $\\Leftrightarrow f(x,y)=f_X(x)f_Y(y)$ on a rectangular support.",
      "tag": "Joint dist"
    },
    {
      "front": "Law of total expectation (tower rule) and conditional-mean fact.",
      "back": "$E[X]=E\\big[E[X\\mid Y]\\big]$ $\\cdot$ $E[g(Y)X\\mid Y]=g(Y)E[X\\mid Y]$ $\\cdot$ $E[X\\mid Y]$ is a random variable (function of $Y$).",
      "tag": "Conditioning"
    },
    {
      "front": "Law of total variance (EVVE decomposition).",
      "back": "$\\operatorname{Var}(Y)=E\\big[\\operatorname{Var}(Y\\mid X)\\big]+\\operatorname{Var}\\big(E[Y\\mid X]\\big)$ = mean of conditional variance + variance of conditional mean.",
      "tag": "Conditioning"
    },
    {
      "front": "Compound / conditional-Poisson mixing moments.",
      "back": "$S=\\sum_{i=1}^{N}X_i$: $E[S]=E[N]E[X]$, $\\operatorname{Var}(S)=E[N]\\operatorname{Var}(X)+(E[X])^{2}\\operatorname{Var}(N)$ $\\cdot$ $N\\mid\\Lambda\\sim\\text{Poisson}(\\Lambda)$: $E[N]=E[\\Lambda]$, $\\operatorname{Var}(N)=E[\\Lambda]+\\operatorname{Var}(\\Lambda)$.",
      "tag": "Conditioning"
    },
    {
      "front": "Sum of $n$ iid: $S=\\sum X_i$ and sample mean $\\bar{X}$.",
      "back": "$E[S]=n\\mu$, $\\operatorname{Var}(S)=n\\sigma^{2}$, $\\operatorname{SD}(S)=\\sigma\\sqrt{n}$ $\\cdot$ $E[\\bar{X}]=\\mu$, $\\operatorname{Var}(\\bar{X})=\\dfrac{\\sigma^{2}}{n}$, $\\operatorname{SD}(\\bar{X})=\\dfrac{\\sigma}{\\sqrt{n}}$.",
      "tag": "Sums/CLT"
    },
    {
      "front": "Closure under independent sums (named distributions).",
      "back": "$\\sum\\text{Bin}(n_i,p)=\\text{Bin}(\\sum n_i,p)$ $\\cdot$ $\\sum\\text{Poisson}(\\lambda_i)=\\text{Poisson}(\\sum\\lambda_i)$ $\\cdot$ $\\sum N(\\mu_i,\\sigma_i^{2})=N(\\sum\\mu_i,\\sum\\sigma_i^{2})$ $\\cdot$ $n$ iid $\\text{Exp}(\\lambda)=\\text{Gamma}(n,\\lambda)$.",
      "tag": "Sums/CLT"
    },
    {
      "front": "Central Limit Theorem (sum and mean forms).",
      "back": "$S_n\\approx N(n\\mu,\\,n\\sigma^{2})$, i.e. $\\dfrac{S_n-n\\mu}{\\sigma\\sqrt{n}}\\to N(0,1)$ $\\cdot$ $\\bar{X}\\approx N\\!\\left(\\mu,\\dfrac{\\sigma^{2}}{n}\\right)$, i.e. $\\dfrac{\\bar{X}-\\mu}{\\sigma/\\sqrt{n}}\\to N(0,1)$.",
      "tag": "Sums/CLT"
    },
    {
      "front": "Continuity correction (normal approx to integer-valued $S$).",
      "back": "Normal approximant $Y\\sim N(E[S],\\operatorname{Var}(S))$: $\\Pr(S\\leq k)\\approx\\Pr(Y\\leq k+0.5)$ $\\cdot$ $\\Pr(S\\geq k)\\approx\\Pr(Y\\geq k-0.5)$ $\\cdot$ $\\Pr(S=k)\\approx\\Pr(k-0.5\\leq Y\\leq k+0.5)$.",
      "tag": "Sums/CLT"
    },
    {
      "front": "Markov and Chebyshev inequalities.",
      "back": "Markov ($X\\geq0$): $\\Pr(X\\geq a)\\leq\\dfrac{E[X]}{a}$ $\\cdot$ Chebyshev: $\\Pr(|X-\\mu|\\geq k\\sigma)\\leq\\dfrac{1}{k^{2}}$.",
      "tag": "Inequalities"
    },
    {
      "front": "Standard normal: table use, symmetry, empirical rule.",
      "back": "$\\Phi(z)=\\Pr(Z\\leq z)$, $\\Phi(-z)=1-\\Phi(z)$ $\\cdot$ percentile: $x_p=\\mu+z_p\\sigma$ $\\cdot$ $68\\%$ within $\\mu\\pm\\sigma$, $95\\%$ within $\\mu\\pm2\\sigma$, $99.7\\%$ within $\\mu\\pm3\\sigma$.",
      "tag": "Inequalities"
    },
    {
      "front": "Bivariate normal: linear combinations and conditional law.",
      "back": "$aX+bY\\sim N\\!\\left(a\\mu_X+b\\mu_Y,\\;a^{2}\\sigma_X^{2}+b^{2}\\sigma_Y^{2}+2ab\\,\\rho\\sigma_X\\sigma_Y\\right)$ $\\cdot$ $E[Y\\mid X{=}x]=\\mu_Y+\\rho\\dfrac{\\sigma_Y}{\\sigma_X}(x-\\mu_X)$, $\\operatorname{Var}(Y\\mid X)=\\sigma_Y^{2}(1-\\rho^{2})$ $\\cdot$ $\\rho=0\\Leftrightarrow$ independent.",
      "tag": "Joint dist"
    },
    {
      "front": "Multinomial (trinomial) counts: marginal and cross-covariance.",
      "back": "$X_i\\sim\\text{Bin}(n,p_i)$: $E[X_i]=np_i$, $\\operatorname{Var}(X_i)=np_i(1-p_i)$ $\\cdot$ $\\operatorname{Cov}(X_i,X_j)=-np_ip_j$, $i\\neq j$ (negative).",
      "tag": "Joint dist"
    },
    {
      "front": "Transformation by Jacobian (1-D, monotone $Y=g(X)$).",
      "back": "$f_Y(y)=f_X\\!\\big(g^{-1}(y)\\big)\\left|\\dfrac{dx}{dy}\\right|$ $\\cdot$ non-monotone: sum over all branches $x_i=g^{-1}(y)$ $\\cdot$ probability integral transform: $F_X(X)\\sim U(0,1)$.",
      "tag": "Transformations"
    },
    {
      "front": "Bivariate Jacobian and convolution for $Z=X+Y$.",
      "back": "$f_{U,V}(u,v)=f_{X,Y}\\big(x(u,v),y(u,v)\\big)|J|$, $J=\\dfrac{\\partial x}{\\partial u}\\dfrac{\\partial y}{\\partial v}-\\dfrac{\\partial x}{\\partial v}\\dfrac{\\partial y}{\\partial u}$ $\\cdot$ independent: $f_Z(z)=\\int f_X(x)f_Y(z-x)\\,dx$.",
      "tag": "Transformations"
    },
    {
      "front": "Order statistics: pdf of max, min, and $k$th (iid, cdf $F$, pdf $f$).",
      "back": "Max: $f_{(n)}(x)=n[F(x)]^{n-1}f(x)$ $\\cdot$ Min: $f_{(1)}(x)=n[1-F(x)]^{n-1}f(x)$ $\\cdot$ $k$th: $f_{(k)}(x)=\\dfrac{n!}{(k-1)!(n-k)!}[F(x)]^{k-1}[1-F(x)]^{n-k}f(x)$.",
      "tag": "Order statistics"
    },
    {
      "front": "Order-statistic cdfs and $U(0,1)$ results.",
      "back": "$F_{(n)}(x)=[F(x)]^{n}$, $F_{(1)}(x)=1-[1-F(x)]^{n}$ $\\cdot$ $U(0,1)$: $X_{(k)}\\sim\\text{Beta}(k,n-k+1)$, $E[X_{(k)}]=\\dfrac{k}{n+1}$.",
      "tag": "Order statistics"
    },
    {
      "front": "Minimum of independent exponentials and which is smallest.",
      "back": "$X_i\\sim\\text{Exp}(\\lambda_i)$ independent: $\\min_i X_i\\sim\\text{Exp}\\!\\left(\\sum_i\\lambda_i\\right)$ $\\cdot$ $\\Pr(X_1=\\min)=\\dfrac{\\lambda_1}{\\sum_i\\lambda_i}$.",
      "tag": "Order statistics"
    },
    {
      "front": "Insurance per-loss payment under deductible $d$: $E[(X-d)_{+}]$.",
      "back": "$E[(X-d)_{+}]=\\int_{d}^{\\infty}S(x)\\,dx=E[X]-E[X\\wedge d]$ $\\cdot$ exponential mean $\\theta$: $E[(X-d)_{+}]=\\theta e^{-d/\\theta}$.",
      "tag": "Insurance layers"
    },
    {
      "front": "Limited expected value $E[X\\wedge u]$ and the layer $d$-to-$u$.",
      "back": "$X\\wedge u=\\min(X,u)$, $E[X\\wedge u]=\\int_{0}^{u}S(x)\\,dx$ $\\cdot$ layer cost $=E[X\\wedge u]-E[X\\wedge d]=\\int_{d}^{u}S(x)\\,dx$ $\\cdot$ coinsurance $\\alpha$ scales by $\\alpha$.",
      "tag": "Insurance layers"
    },
    {
      "front": "Per-loss vs per-payment under deductible $d$.",
      "back": "Per loss: $E[(X-d)_{+}]$ (averages over all losses, including $0$-pay) $\\cdot$ per payment: $E[X-d\\mid X>d]=\\dfrac{E[(X-d)_{+}]}{S(d)}$ (larger).",
      "tag": "Insurance layers"
    }
  ]
}