← Probability & Distributions
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Probability & Distributions

Chapter 2 · Distributions

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Binomial
X∼B(n,p):  P(X=x)=(nx)px(1−p)n−xX\sim B(n,p):\; P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}
Binomial mean / variance
E(X)=np,Var(X)=np(1−p)E(X)=np,\quad \mathrm{Var}(X)=np(1-p)
Normal
X∼N(μ,σ2)X\sim N(\mu,\sigma^2)
Symmetric bell about μ\mu; total area = 1.
Standardisation
Z=X−μσ∼N(0,1)Z=\frac{X-\mu}{\sigma}\sim N(0,1)
Empirical rule
≈68% within μ±σ\mu\pm\sigma, ≈95% within ±2σ\pm2\sigma, ≈99.7% within ±3σ\pm3\sigma.
Inverse normal
x=μ+zσx=\mu+z\sigma
Find zz from the required probability, then scale back.
Binomial
X∼B(n,p),  E(X)=np,  Var=np(1−p)X\sim B(n,p),\; E(X)=np,\; \text{Var}=np(1-p)
Fixed n trials, constant p, independent.
Normal
Symmetric bell; use GDC for P(a
Expected value
E(X)=∑x P(x)E(X)=\sum x\,P(x)
Common trap
Binomial is discrete (use =), normal is continuous (P(X=x)=0); inverse-normal for ‘find x given probability’.