2026-08-07-Preview

2026-08-07-Preview · Probability Theory Ch4 Random Variables（随机变量）

Ref: Lecture slides Ch4 · Ross Ch4 Random Variables

Key Concepts

Term	Definition
Random variable（随机变量）	A real-valued function defined on the sample space
pmf（概率质量函数）	$p(a)=P(X=a)$; ${\sum }_{i}p(x_i)=1$ for a discrete r.v.
CDF（累积分布函数）	$F(a)=P(X\le a)={\sum }_{x\le a}p(x)$
Expectation（期望）	$E(X)={\sum }_i{x}_{i}p(x_i)$; $E[aX+b]=aE[X]+b$
Variance（方差）	$Var(X)=E[(X-\mu {)}^2]=E[X^2]-{\mu }^2$; $Var(aX+b)=a^{2}Var(X)$
Bernoulli / Binomial（伯努利 / 二项）	$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$; $E=np$, $Var=np(1-p)$
Poisson（泊松）	$P(X=i)=e^{-\lambda }{\lambda }^i/i!$; $E=Var=\lambda$; approximates Binomial when $n$ large, $p$ small
Geometric / Negative Binomial（几何 / 负二项）	$P(X=n)=(1-p{)}^{n-1}p$; $E=1/p$, $Var=(1-p)/p^2$; neg. bin. $E=r/p$, $Var=r(1-p)/p^2$

Per-Subsection: Definition & Examples

4.1 Random Variables（随机变量）

Definition

A random variable is a real-valued function on the sample space: we are interested in a function of the outcome (e.g. the sum of two dice, the number of heads) rather than the outcome itself.

Example: exam until you pass（考试直到通过）— pass with proba $p$ each independent trial; $X$ = number of exams needed.

• $P(X=n)=(1-p{)}^{n-1}p$;
• ${\sum }_{k=1}^{\infty }(1-p{)}^{k-1}p=1$ — probabilities sum to 1, a valid distribution.

4.2 Discrete Random Variables（离散随机变量）

Definition

$X$ is discrete if it takes at most countably many values. Its pmf is $p(a)=P(X=a)$, with $p(x_i)\ge 0$ and ${\sum }_{i}p(x_i)=1$. Its CDF is $F(a)=P(X\le a)={\sum }_{x\le a}p(x)$.

Example: $p(1)=1/4,p(2)=1/2,p(3)=1/8,p(4)=1/8$.

• CDF is a step function: $F(1)=1/4$, $F(2)=3/4$, $F(3)=7/8$, $F(4)=1$;
• If $p(i)=c{\lambda }^i/i!$, then $c=e^{-\lambda }$ (normalization), $P(X=0)=e^{-\lambda }$, $P(X>2)=1-e^{-\lambda }(1+\lambda +{\lambda }^2/2)$.

4.3 Expected Value（期望）

Definition

$E(X)={\sum }_i{x}_{i}p(x_i)$ — the weighted average of the possible values, weighted by probability.

Example: asking someone out（要不要表白）— happiness $+1000$ if accepted (proba 0.1), $-100$ if rejected.

• $E=0.1\times 1000+0.9\times (-100)=10>0$ → expected happiness is positive, ask them out；
• fair die: $E(X)=1\times \frac{1}{6}+\cdots +6\times \frac{1}{6}=\frac{7}{2}$.

4.4 Expectation of a Function（函数的期望）

Definition

If $Y=g(X)$, then $E(Y)=E(g(X))={\sum }_{i}g(x_i)p_X(x_i)$.

Corollary: $E[aX+b]=aE[X]+b$.

Example: $X$ uniform on $\{-2,-1,0,1,2\}$, $Y=X^2$:

• $E(Y)=\frac{4+1+0+1+4}{5}=2$ — compute via pmf of $X$, not by guessing the pmf of $Y$.

4.5 Variance（方差）

Definition

$Var(X)=E\left[(X-\mu {)}^2\right]=E[X^2]-{\mu }^2$; $Std(X)=\sqrt{Var(X)}$; $Var(aX+b)=a^{2}Var(X)$.

Example: fair die — $E(X)=\frac{7}{2}$, $E(X^2)=\frac{91}{6}$, so

$$Var(X)=\frac{91}{6}-{\left(\frac{7}{2}\right)}^2=\frac{35}{12}$$

4.6 Bernoulli & Binomial（伯努利与二项分布）

Definition

Bernoulli: $X=1$ (success) w.p. $p$, $X=0$ w.p. $1-p$; $E=p$, $Var=p(1-p)$.

Binomial: $X=X_1+\cdots +X_n$ counts successes in $n$ independent trials:

$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k},E(X)=np,Var(X)=np(1-p)$

Example: friend claims to taste Pepsi vs Coke（盲测）— 5 trials, 4 correct. If guessing randomly, $X\sim Bin(5,1/2)$:

• $P(X\ge 4)=P(X=4)+P(X=5)=0.1875$;
• Not convincing evidence — 18.75% chance of getting 4+ right by luck（统计显著性直觉）.

4.7 Poisson（泊松分布）

Definition

$P(X=i)=e^{-\lambda }\frac{{\lambda }^i}{i!}$ ($i=0,1,2,\dots$); $E(X)=Var(X)=\lambda$.

Poisson ≈ Binomial when $n$ is large and $p$ small with $\lambda =np$ — “rare events in many trials”.

Example: 50 misprints in a 250-page book（书中的错字）— $\lambda =50/250=0.2$ per page.

• $P(\text{no misprint on page 100})=e^{-0.2}=0.819$;
• $P(2\text{ misprints})=e^{-0.2}\frac{0.2^2}{2!}=0.0164$.

4.8 Geometric & Negative Binomial（几何与负二项分布）

Definition

Geometric: number of trials until first success, $P(X=n)=(1-p{)}^{n-1}p$; memoryless property $P(X>s|X>t)=P(X>s-t)$; $E=1/p$, $Var=(1-p)/p^2$.

Negative binomial: trials until $r$-th success, $P(X=n)=\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)p^r(1-p{)}^{n-r}$; $E=r/p$, $Var=r(1-p)/p^2$.

Example: collecting $n$ pictures（集齐球星卡）— $Y=X_1+\cdots +X_n$ where $p_k=1-\frac{k-1}{n}$:

• $E(Y)={\sum }_k\frac{1}{p_k}=n\left(1+\frac{1}{2}+\cdots +\frac{1}{n}\right)$;
• $n=6$: $E(Y)=14.7$; $n=14$: $E(Y)=45.5$ — collecting takes much longer than intuition suggests.