Probability & Random Variables

Measure-Theoretic Foundations

Probability assigns a real number $P(E)\in [0,1]$ to events in a sample space $\mathrm{\Omega }$ satisfying Kolmogorov's axioms: $P(\mathrm{\Omega })=1$ and countable additivity for mutually disjoint events $P(\bigcup E_i)=\sum P(E_i)$.

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1. Conditional Probability & Multiplication Rule

📊Visual Mathematical Intuition

Sample Space Partition & Bayesian Conditional Branching

Total Probability:$P(A) = \sum_{i=1}^n P(A|B_i) P(B_i)$

Bayes Theorem:$P(B_k|A) = \frac{P(A|B_k) P(B_k)}{\sum P(A|B_i) P(B_i)}$

Sample space partition $S = \bigcup B_i$ intersecting target event $A$, alongside the Bayesian conditional probability decision tree showing prior and posterior probabilities.

$$P(A|B)=\frac{P(A\cap B)}{P(B)}(P(B)>0)$$

• Independent Events: $A$ and $B$ are independent iff $P(A\cap B)=P(A)\cdot P(B)⟺P(A|B)=P(A)$.

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2. Law of Total Probability & Bayes' Theorem

Bayes' Inverse Probability Theorem

Let $E_1,E_2,\dots ,E_n$ form a partition of the sample space $\mathrm{\Omega }$ ($E_i\cap E_j=\mathrm{\varnothing }$ and $\bigcup E_i=\mathrm{\Omega }$) with $P(E_i)>0$. For any event $A$ with $P(A)>0$:

$$P(E_k|A)=\frac{P(E_k)\cdot P(A|E_k)}{\sum _{i=1}^{n}P(E_i)\cdot P(A|E_i)}$$

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3. Random Variables, Expectation & Variance

For a discrete random variable $X$ with probability mass function $P(X=x_i)=p_i$:

• Expectation (Mean): $E[X]=\mu =\sum x_i{p}_i$
• Variance: $\text{Var}(X)={\sigma }^2=E[(X-\mu {)}^2]=E[X^2]-(E[X]{)}^2$

Properties of Expectation and Variance

1. $E[aX+b]=aE[X]+b$
2. $\text{Var}(aX+b)=a^2\text{Var}(X)$
3. For independent random variables $X$ and $Y$: $\text{Var}(X\pm Y)=\text{Var}(X)+\text{Var}(Y)$.

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4. Binomial Distribution $B(n,p)$

For $n$ independent Bernoulli trials with success probability $p$ and failure probability $q=1-p$:

$$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k{q}^{n-k},k=0,1,2,\dots ,n$$

• Mean: $E[X]=np$
• Variance: $\text{Var}(X)=npq$
• Standard Deviation: $\sigma =\sqrt{npq}$