#set document(title: "5.8 Chapter 5 Formulas", author: "Rachel Webb") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 5.8#h(0.6em)Chapter 5 Formulas #figure(table( columns: 2, align: left, inset: 6pt, table.header([#strong[Discrete Distribution Table:] 0 ≤ P(x#sub[i]) ≤ 1 ∑ P(x#sub[i]) = 1], [#strong[Discrete Distribution Mean:] μ = Σ(x#sub[i] ∙ P(x#sub[i]))]), [#strong[Discrete Distribution Variance:] σ#super[2] = ∑(x#sub[i]#super[2] ∙P(x#sub[i])) – μ#super[2]], [#strong[Discrete Distribution Standard Deviation:] σ = #math.equation(block: false, alt: "the square root of σ squared")[$sqrt(σ^(2))$]], [#strong[Geometric Distribution:] P(X = x) = #emph[p]∙ #emph[q] #super[(x – 1)] , x = 1, 2, 3, …], [#strong[Geometric Distribution Mean:] μ = #math.equation(block: false, alt: "the fraction 1 over p")[$frac(1, p)$] Variance: σ#super[2] = #math.equation(block: false, alt: "the fraction 1 − p over p squared")[$frac(1 − p, p^(2))$] Standard Deviation: σ = #math.equation(block: false, alt: "the square root of the fraction 1 minus p over p squared")[$sqrt(frac(1 − p, p^(2)))$]], [#strong[Binomial Distribution:] P(X = x) = #sub[n]C#sub[x]·p#super[x] ·q#super[(n-x]) , x = 0, 1, 2, … , n], [#strong[Binomial Distribution Mean:] μ = n ∙ p Variance: σ 2 = n ∙ p ∙ q Standard Deviation: σ = #math.equation(block: false, alt: "the square root of n times p times q")[$sqrt(n) · p · q$]], [#strong[Hypergeometric Distribution:] P(X = x) = #math.equation(block: false, alt: "the fraction a C sub x times b C sub n minus x over N C sub n")[$frac(a C_(x) · b C_(n − x), N C_(n))$]], [p = P(success) q = P(failure) = 1 – p n = sample size N = population size], [#strong[Unit Change for Poisson Distribution:] New μ = old μ(#math.equation(block: false, alt: "the fraction new units over old units")[$frac(" new units ", " old units ")$])], [#strong[Poisson Distribution:] P(X = x) = #math.equation(block: false, alt: "the fraction e to the power minus μ μ to the power x over x !")[$frac(e^(− μ) μ^(x), x !)$]], ))