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probability

Probability laws, conditional & Bayes, discrete & continuous random variables, binomial & normal.

probability laws · discrete random variables · continuous random variables · binomial · normal

Probability laws

Probability laws. \(P(A)=\dfrac{n(A)}{n(U)}\); \(0\le P(A)\le1\); \(P(A')=1-P(A)\). \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Mutually exclusive: \(P(A\cap B)=0\). Conditional: \(P(A|B)=\dfrac{P(A\cap B)}{P(B)}\), \(P(A\cap B)=P(B)P(A|B)=P(A)P(B|A)\). Independent: \(P(A\cap B)=P(A)P(B)\), equivalently \(P(A|B)=P(A)\). Tree diagrams: multiplication along branches, addition across disjoint branches. With replacement usually independent; without replacement usually conditional.

Bayes. For partition \(A_1,\ldots,A_n\): \(P(B)=\sum_i P(A_i)P(B|A_i)\), \(P(A_j|B)=\dfrac{P(A_j)P(B|A_j)}{\sum_i P(A_i)P(B|A_i)}\). Two-event form: \(P(A|B)=\dfrac{P(A)P(B|A)}{P(A)P(B|A)+P(A')P(B|A')}\).

Discrete random variables

Discrete random variables. \(\sum p(x_i)=1\). \(\mu=\E(X)=\sum x_i p(x_i)\); \(\E(g(X))=\sum g(x_i)p(x_i)\). \(\Var(X)=\sum(x_i-\mu)^2p(x_i)=\E(X^2)-[\E(X)]^2\); \(\sigma=\sqrt{\Var(X)}\). \(\E(aX+b)=a\E(X)+b\), \(\Var(aX+b)=a^2\Var(X)\). Fair game: \(\E(\text{gain})=0\).

Continuous random variables

Continuous random variables. PDF: \(f(x)\ge0\), \(\int_{-\infty}^{\infty}f(x)\dd x=1\). \(P(a\lt X\lt b)=\int_a^b f(x)\dd x\); point probability \(P(X=c)=0\). CDF: \(F(x)=P(X\le x)=\int_{-\infty}^{x}f(t)\dd t\), \(f=F'\). Median \(m\): \(F(m)=1/2\). Mode: \(f\) maximum. \(\mu=\E(X)=\int_{-\infty}^{\infty}xf(x)\dd x\); \(\E(g(X))=\int g(x)f(x)\dd x\). \(\Var(X)=\int(x-\mu)^2f(x)\dd x=\E(X^2)-\mu^2\). Linear transform rules same as discrete.

Binomial distribution

Binomial. Conditions: fixed \(n\), independent trials, two outcomes, constant success probability \(p\). \(X\sim B(n,p)\), \(q=1-p\); \(P(X=x)=\binom nx p^xq^{n-x}\), \(x=0,\ldots,n\). \(\E(X)=np\), \(\Var(X)=npq\), \(\sigma=\sqrt{npq}\). Cumulative: \(P(X\le k)\); \(P(X\ge k)=1-P(X\le k-1)\); \(P(a\le X\le b)=P(X\le b)-P(X\le a-1)\).

Normal distribution

Normal. \(X\sim N(\mu,\sigma^2)\). Standardization: \(Z=\dfrac{X-\mu}{\sigma}\sim N(0,1)\), \(x=\mu+z\sigma\). \(P(a\lt X\lt b)=\Phi\!\left(\frac{b-\mu}{\sigma}\right)-\Phi\!\left(\frac{a-\mu}{\sigma}\right)\). Symmetry: \(P(X\lt\mu-a)=P(X\gt\mu+a)\). Empirical: \(\mu\pm\sigma\approx68\%\), \(\mu\pm2\sigma\approx95\%\), \(\mu\pm3\sigma\approx99.7\%\). Unknown \(\mu,\sigma\): from percentiles \(x_i=\mu+z_i\sigma\), solve simultaneous equations.