2026-08-11-Preview

2026-08-11-Preview · Probability Theory Ch5 Continuous Random Variables（连续型随机变量）

Ref: Lecture slides Ch5 · Ross Ch5 Continuous Random Variables

Key Concepts

Term	Definition
Continuous r.v. / pdf（连续型随机变量 / 概率密度）	$P(X\in A)={\int }_{A}f(x)dx$; pdf $f\ge 0$; cdf $F(x)={\int }_{-\infty }^{x}f(y)dy$, $f=F^{\prime }$
Density is not probability（密度不是概率）	$P(X=x)=0$ for every $x$; $f(x)$ may exceed 1; $<$ vs $\le$ at endpoints does not matter
Expectation（期望）	$E(X)={\int }_{-\infty }^{\infty }xf(x)dx$; $E(g(X))=\int g(x)f(x)dx$
Variance（方差）	$Var(X)=E[(X-\mu {)}^2]=E[X^2]-{\mu }^2$
Quantile / Median（分位数 / 中位数）	$F(Q(\alpha ))=\alpha$; median $m$ solves ${\int }_{-\infty }^{m}f={\int }_m^{\infty }f=1/2$
Uniform（均匀分布）	$f(x)=1/(d-c)$ on $(c,d)$; $E=(c+d)/2$, $Var=(d-c{)}^2/12$
Exponential（指数分布）	$f(x)=\lambda e^{-\lambda x}$, $x\ge 0$; $E=1/\lambda$, $Var=1/{\lambda }^2$; memoryless; constant hazard $\lambda$
Normal / standard normal（正态 / 标准正态）	$f(x)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-(x-\mu {)}^2/2{\sigma }^2}$; $Z=(X-\mu )/\sigma$; 68-95-99.7 rule
Log-normal / Gamma / Beta（对数正态 / 伽马 / 贝塔）	$\log X\sim N(\mu ,{\sigma }^2)$; $\Gamma (\alpha +1)=\alpha \Gamma (\alpha )$, $\Gamma (1/2)=\sqrt{\pi }$; $B(\alpha ,\beta )=\Gamma (\alpha )\Gamma (\beta )/\Gamma (\alpha +\beta )$; $U(0,1)=Beta(1,1)$
Function of a r.v.（随机变量函数）	monotone $g$: $f_Y(y)=f_X(g^{-1}(y)),

Per-Subsection: Definition & Examples

5.1 Continuous Random Variables（连续型随机变量）

Definition

$X$ is a continuous r.v. if there exists a nonnegative function $f:\mathbb{R}\to [0,\infty )$ such that for any set $A\subset \mathbb{R}$,

$P(X\in A)={\int }_{A}f(x)dx$

$f$ is the probability density function (pdf); the cdf is $F(x)=P(X\le x)={\int }_{-\infty }^{x}f(y)dy$, so $f(x)=F^{\prime }(x)$, and ${\int }_{\mathbb{R}}f(x)dx=1$.

Definition (density is not probability)

• $f(x)$ is not a probability: $f(1)\ne P(X=1)$; it is a density, so $f(x)>1$ is allowed.
• For a continuous r.v., $P(X=x)=0$ for every $x$, hence $P(a\le X\le b)=P(a<X<b)$ — endpoints do not matter.
• Keep track of the support (range) of $f$ (e.g. $[0,1]$ or $[0,\infty )$) when integrating.

Example: toy density（分段密度定常数）— $f(x)=Cx$ on $[0,1]$, $C{x}^2$ on $[-1,0]$, $0$ otherwise.

• Normalization: $\int f=C\left({\int }_{-1}^0{x}^{2}dx+{\int }_0^{1}xdx\right)=C(1/3+1/2)=5C/6=1$ ⇒ $C=6/5$;
• $P(X<0.5)=\frac{6}{5}\left({\int }_{-1}^0{x}^{2}dx+{\int }_0^{0.5}xdx\right)=\frac{6}{5}\left(\frac{1}{3}+\frac{1}{8}\right)=\frac{11}{20}=0.55$.

Example: insurance policy（理赔额的条件概率）— $V=100000X$, $f(x)=c(1-x{)}^4$ on $(0,1)$.

• $c=5$; cdf $F_X(x)=1-(1-x{)}^5$ on $[0,1]$;
• $P(V>40000\mid V>10000)=\frac{P(X>0.4)}{P(X>0.1)}=\frac{(0.6{)}^5}{(0.9{)}^5}=(2/3{)}^5\approx 0.132$.

5.2 (i) Expectation（期望）

Definition

$E(X)={\int }_{-\infty }^{\infty }xf(x)dx$; more generally $E(g(X))={\int }_{-\infty }^{\infty }g(x)f(x)dx$.

• Uniform on $(c,d)$: $E(X)=(c+d)/2$;
• Exponential($\lambda$): $E(X)=1/\lambda$ (integration by parts);
• Symmetric (even) density: $E(X)=0$.

Example: heavy-tailed density（厚尾分布期望不存在）— $f(x)=1/(x+1{)}^2$, $x\ge 0$.

• $E(X+1)={\int }_0^{\infty }\frac{x+1}{(x+1{)}^2}dx={\int }_0^{\infty }\frac{dx}{x+1}=\infty$;
• $E(X)$ does not exist; such heavy-tailed distributions appear in finance and actuarial science.

Example: selling printers（打印机退款期望）— lifetime $T\sim$ Exponential(mean 2), full refund \200$iffailsinyear1,halfrefund$$100$ in year 2.

• $P(T<1)=1-e^{-1/2}=0.393$; $P(1<T<2)=e^{-1/2}-e^{-1}=0.239$;
• per printer $E(X_i)=200\times 0.393+100\times 0.239=102.56$; for 100 printers $E(\sum X_i)=100\times 102.56=10256$（期望可加，无需独立性）.

Example: failure discovery（故障发现时间）— $T\sim$ Exponential(1/3), observed $X=\max (T,2)$.

• $E(X)={\int }_0^{2}2\cdot \frac{1}{3}{e}^{-t/3}dt+{\int }_2^{\infty }t\cdot \frac{1}{3}{e}^{-t/3}dt=2+3e^{-2/3}\approx 3.54$.

Remark

$E(X)=\arg {\min }_{a}E[(X-a{)}^2]$: the mean minimizes the mean squared error (MSE).

5.2 (ii) Variance（方差）

Definition

$Var(X)=E[(X-E(X){)}^2]=E[X^2]-E[X{]}^2$; $\sigma =SD(X)=\sqrt{Var(X)}$.

• Uniform on $(c,d)$: $E(X^2)=(c^2+d^2+cd)/3$, $Var(X)=(d-c{)}^2/12$;
• Exponential($\lambda$): $E(X^2)=2/{\lambda }^2$, $Var(X)=1/{\lambda }^2$.

Example: repair cost and insurance payment（免赔额下的赔付标准差）— repair cost $X\sim U(0,1500)$, deductible 250, payment $Y=\max (X-250,0)$.

• $E(Y)={\int }_{250}^{1500}\frac{x-250}{1500}dx=521$;
• $E(Y^2)={\int }_{250}^{1500}\frac{(x-250{)}^2}{1500}dx=434028$;
• $SD(Y)=\sqrt{434028-521^2}\approx 403$.

5.2 (iii) Quantiles and Median（分位数与中位数）

Definition

The median $m$ satisfies ${\int }_{-\infty }^{m}f(x)dx={\int }_m^{\infty }f(x)dx=1/2$, i.e. $P(X\le m)=P(X\ge m)=1/2$.

The $\alpha$-quantile $Q(\alpha )$ satisfies $F_X(Q(\alpha ))=\alpha$; $Q(0.25)$, $Q(0.5)$, $Q(0.75)$ are the lower quartile, median and upper quartile (boxplot).

• Uniform: median $=(c+d)/2$, the same as the mean;
• Exponential: $e^{-\lambda m}=1/2$ ⇒ median $=\ln 2/\lambda$, different from the mean $1/\lambda$;
• For symmetric distributions (e.g. normal), mean = median; for skewed ones they differ.

Remark

Median minimizes the mean absolute error: $Q(0.5)=\arg {\min }_{a}E|X-a|$, while the mean minimizes the mean squared error — median is more robust to outliers.

5.3 Uniform Distribution（均匀分布）

Definition

(slides refer back to 5.1) $X\sim U(c,d)$: $f(x)=\frac{1}{d-c}$ on $(c,d)$; cdf $F(x)=\frac{x-c}{d-c}$; $E(X)=\frac{c+d}{2}$, $Var(X)=\frac{(d-c{)}^2}{12}$.

5.4 Normal Distribution（正态分布）

Definition

$X\sim N(\mu ,{\sigma }^2)$: $f(x)=\frac{1}{\sqrt{2\pi }\sigma }\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right)$, $-\infty <x<\infty$; $E(X)=\mu$, $Var(X)={\sigma }^2$.

Definition (68-95-99.7 rule)

$P(\mu -\sigma \le X\le \mu +\sigma )\approx 0.68$; $P(\mu -2\sigma \le X\le \mu +2\sigma )\approx 0.95$; $P(\mu -3\sigma \le X\le \mu +3\sigma )\approx 0.997$.

Example: SAT scores（经验法则的快速估算）— verbal scores $\approx N(504,111^2)$: 95% of scores lie in $[504\pm 222]=[282,726]$.

Definition (standardizing)

If $X\sim N(\mu ,{\sigma }^2)$, then $Z=(X-\mu )/\sigma \sim N(0,1)$, with cdf $\Phi (z)=P(Z\le z)$ and $\Phi (-z)=1-\Phi (z)$:

$P(a\le X\le b)=\Phi \left(\frac{b-\mu }{\sigma }\right)-\Phi \left(\frac{a-\mu }{\sigma }\right)$

Linear transformation preserves normality: $Y=aX+b\sim N(a\mu +b,a^2{\sigma }^2)$.

Example: normal computations（正态查表）— $X\sim N(2,25)$.

• (a) $P(1\le X\le 4)=\Phi (0.4)-\Phi (-0.2)=\Phi (0.4)+\Phi (0.2)-1$;
• (b) $P(X>0)=P(Z>-0.4)=\Phi (0.4)$;
• (c) $P((X-2{)}^2>5)=P(|Z|>1/\sqrt{5})=2\left(1-\Phi (1/\sqrt{5})\right)$.

Example: signal transmission（二元信号传输）— send 2 for message 1, $-2$ for message 0; received $R=x+N$ with $N\sim N(0,1)$; decode “1” if $R\ge 0.5$.

• transmit 0: $R=-2+N$, $P(\text{correct})=P(R<0.5)=\Phi (2.5)\approx 0.994$;
• transmit 1: $R=2+N$, $P(\text{correct})=P(R>0.5)=\Phi (1.5)\approx 0.933$;
• Key idea: both error probabilities are just standardized tail probabilities; the threshold 0.5 trades off the two types of error.

Example: threshold signal（阈值截断信号的期望）— $Y=X$ if $X\ge m$, otherwise $0$.

• $E(Y)={\int }_m^{\infty }\frac{x}{\sqrt{2\pi }\sigma }{e}^{-(x-\mu {)}^2/2{\sigma }^2}dx$
• $=\frac{\sigma }{\sqrt{2\pi }}\exp \left(-\frac{(m-\mu {)}^2}{2{\sigma }^2}\right)+\mu \left(1-\Phi \left(\frac{m-\mu }{\sigma }\right)\right)$;
• Key idea: split $x=(x-\mu )+\mu$; the first term is an exact Gaussian integral, the second is a tail probability.

Definition (normal approximation to binomial)

For $X\sim Bin(n,p)$, as $n\to \infty$: $X\approx N(np,np(1-p))$, hence

$P(a\le X\le b)\approx \Phi \left(\frac{b-np}{\sqrt{np(1-p)}}\right)-\Phi \left(\frac{a-np}{\sqrt{np(1-p)}}\right)$

Example: color blindness（二项分布的正态近似）— $X$ = number of color-blind men among 818, $p=0.1$: $\mu =81.8$, $\sigma =8.5802$.

• $P(X\le 100)\approx \Phi (2.12)-\Phi (-9.53)\approx 0.983$.

5.5 Exponential Distribution（指数分布）

Definition

Interarrival times of a Poisson process with rate $\lambda$: $f(x)=\lambda e^{-\lambda x}$, $x\ge 0$; $E(X)=1/\lambda$, $Var(X)=1/{\lambda }^2$.

Memoryless property: $P(X>s+t\mid X>s)=P(X>t)$ — the exponential is the only continuous memoryless distribution.

Example: phone calls（电话等待时间）— calls arrive at rate 2/hour, $X$ = waiting time (hours) until the next call.

• (a) $P(X>1)=e^{-2}\approx 0.13$;
• (b) $P(X<1/6)=1-e^{-1/3}\approx 0.28$;
• (c) $P(1/6<X<1)=e^{-1/3}-e^{-2}\approx 0.58$.

Example: nuclear power plant（三个独立安全系统）— lifetimes $X_i\sim$ Exp(1), Exp(0.5), Exp(0.1); never inspected for 5 years.

• $P(X_i<5)=1-e^{-5{\lambda }_i}$: 0.9933, 0.9179, 0.3935;
• $P(\text{no system works})=0.9933\times 0.9179\times 0.3935\approx 0.3588$.

Definition (hazard function)

For a lifetime r.v. with density $f$ and cdf $F$, the hazard rate is $\lambda (t)=f(t)/(1-F(t))$, the conditional failure intensity:

$P(X\in (t,t+dt)\mid X>t)\approx \lambda (t)dt$

For Exponential($\lambda$): $\lambda (t)=\lambda e^{-\lambda t}/e^{-\lambda t}=\lambda$ — constant failure rate; the hazard function uniquely determines $F$.

5.6 Other Continuous Distributions（其他连续型分布）

Definition (log-normal)

$X$ is log-normal($\mu ,{\sigma }^2$) if $f_X(x)=\frac{1}{\sqrt{2\pi }\sigma x}\exp \left(-\frac{(\log x-\mu {)}^2}{2{\sigma }^2}\right)$, $x>0$; equivalently $\log X\sim N(\mu ,{\sigma }^2)$.

Finance: geometric Brownian motion $d{P}_t=\mu P_{t}dt+\sigma P_{t}d{W}_t$; by Itô’s lemma $\log P_t$ is normal, so $P_t$ is log-normal.

Definition (gamma)

$X\sim Gamma(\alpha ,\lambda )$: $f_X(x)=\frac{{\lambda }^{\alpha }}{\Gamma (\alpha )}{x}^{\alpha -1}{e}^{-\lambda x}$, $x>0$, where $\Gamma (\alpha )={\int }_0^{\infty }{t}^{\alpha -1}{e}^{-t}dt$.

$\Gamma (\alpha +1)=\alpha \Gamma (\alpha )$; $\Gamma (k)=(k-1)!$ for integer $k$; $\Gamma (1/2)=\sqrt{\pi }$; $Gamma(1,\lambda )=Exp(\lambda )$; $\alpha$ is the shape parameter, $\lambda$ the rate; $E(X)=\alpha /\lambda$, $Var(X)=\alpha /{\lambda }^2$.

Definition (beta)

$X\sim Beta(\alpha ,\beta )$: $f_X(x)=\frac{1}{B(\alpha ,\beta )}{x}^{\alpha -1}(1-x{)}^{\beta -1}$, $0<x<1$, with $B(\alpha ,\beta )={\int }_0^1{x}^{\alpha -1}(1-x{)}^{\beta -1}dx=\frac{\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )}$.

Support is $[0,1]$; $U(0,1)=Beta(1,1)$.

5.7 Distribution of a Function of a Random Variable（随机变量函数的分布）

Definition

For $Y=g(X)$: CDF method $F_Y(y)=P(g(X)\le y)$, then differentiate.

If $g$ is strictly monotone and differentiable:

$f_Y(y)=f_X(g^{-1}(y))\left|\frac{d}{dy}{g}^{-1}(y)\right|$

Linear case $Y=aX+b$ ($a>0$): $f_Y(y)=\frac{1}{a}{f}_X\left(\frac{y-b}{a}\right)$.

Proof

• Increasing $g$: $F_Y(y)=P(X\le g^{-1}(y))=F_X(g^{-1}(y))$; chain rule gives $f_Y(y)=f_X(g^{-1}(y))(g^{-1}{)}^{\prime }(y)$;
• decreasing $g$: the inequality flips, which produces the absolute value $|(g^{-1}{)}^{\prime }(y)|$.

Example: Celsius to Fahrenheit（线性变换）— $Y=\frac{9}{5}X+32$: $f_Y(y)=\frac{5}{9}{f}_X\left(\frac{5}{9}(y-32)\right)$.

Example: polar to Cartesian（极坐标转直角坐标）— $\theta \sim U(0,\pi /2)$, $Y=\sin \theta$.

• For $0\le y\le 1$: $F_Y(y)=P(\sin \theta \le y)=P(\theta \le {\sin }^{-1}y)=\frac{2}{\pi }{\sin }^{-1}y$;
• $f_Y(y)=\frac{2}{\pi \sqrt{1-y^2}}$ on $(0,1)$ — the arcsine density.

Example: squared variable（平方变换）— $Y=X^2$: for $y\ge 0$,

$f_Y(y)=\frac{1}{2\sqrt{y}}\left(f_X(\sqrt{y})+f_X(-\sqrt{y})\right)$

Example: investment（投资终值的分布）— $R\sim U(0.04,0.08)$, $V=10000e^R$.

• $F_V(v)=P(R\le \ln (v/10000))=25\ln (v/10000)-1$;
• $f_V(v)=25/v$ (monotone formula with $g^{-1}(v)=\ln (v/10000)$, or $F_V^{\prime }$).

Example: Cauchy reciprocal（柯西分布取倒数不变）— $f_X(x)=\frac{1}{\pi (1+x^2)}$, $Y=1/X$.

• $g^{-1}(y)=1/y$, $|(g^{-1}{)}^{\prime }(y)|=1/y^2$;
• $f_Y(y)=\frac{1}{\pi (1+1/y^2)}\cdot \frac{1}{y^2}=\frac{1}{\pi (1+y^2)}=f_X(y)$ — Cauchy is invariant under reciprocation.