#set document(title: "6.7 Chapter 6 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")) == 6.7#h(0.6em)Chapter 6 Formulas #figure(table( columns: 2, align: left, inset: 6pt, table.header([#strong[Uniform Distribution] #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over b minus a , for a less than or equal to x less than or equal to b")[$f ( x ) = frac(1, b − a) , " for " a ≤ x ≤ b$] #math.equation(block: false, alt: "P open parenthesis X greater than or equal to x close parenthesis equals P open parenthesis X greater than x close parenthesis equals open parenthesis the fraction 1 over b minus a close parenthesis times open parenthesis b minus x close parenthesis")[$P ( X ≥ x ) = P ( X > x ) = ( frac(1, b − a) ) · ( b − x )$] #math.equation(block: false, alt: "P open parenthesis X less than or equal to x close parenthesis equals P open parenthesis X less than x close parenthesis equals open parenthesis the fraction 1 over b minus a close parenthesis times open parenthesis x minus a close parenthesis")[$P ( X ≤ x ) = P ( X < x ) = ( frac(1, b − a) ) · ( x − a )$] #math.equation(block: false, alt: "P open parenthesis x sub 1 less than or equal to X less than or equal to x sub 2 close parenthesis equals P open parenthesis x sub 1 less than X less than x sub 2 close parenthesis equals open parenthesis the fraction 1 over b minus a close parenthesis times open parenthesis x sub 2 minus x sub 1 close parenthesis")[$P ( x_(1) ≤ X ≤ x_(2) ) = P ( x_(1) < X < x_(2) ) = ( frac(1, b − a) ) · ( x_(2) − x_(1) )$]], [#strong[Exponential Distribution] #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over μ e to the power open parenthesis minus the fraction x over μ close parenthesis , for x greater than or equal to 0")[$f ( x ) = frac(1, μ) e^(( − frac(x, μ) )) , " for " x ≥ 0$] #math.equation(block: false, alt: "P open parenthesis X greater than or equal to x close parenthesis equals P open parenthesis X greater than x close parenthesis equals e to the power minus x / μ")[$P ( X ≥ x ) = P ( X > x ) = e^(− x / μ)$] #math.equation(block: false, alt: "P open parenthesis X less than or equal to x close parenthesis equals P open parenthesis X less than x close parenthesis equals 1 minus e to the power minus x / μ")[$P ( X ≤ x ) = P ( X < x ) = 1 − e^(− x / μ)$] #math.equation(block: false, alt: "P open parenthesis x sub 1 less than or equal to X less than or equal to x sub 2 close parenthesis equals P open parenthesis x sub 1 less than X less than x sub 2 close parenthesis equals e to the power open parenthesis minus the fraction x sub 1 over μ close parenthesis minus e to the power open parenthesis minus the fraction x sub 2 over μ close parenthesis")[$P ( x_(1) ≤ X ≤ x_(2) ) = P ( x_(1) < X < x_(2) ) = e^(( − frac(x_(1), μ) )) − e^(( − frac(x_(2), μ) ))$]]), [#strong[Standard Normal Distribution] #math.equation(block: false, alt: "μ equals 0 , σ equals 1")[$μ = 0 , σ = 1$] #linebreak() #math.equation(block: false, alt: "z -score: z equals the fraction x minus μ over σ")[$z "-score: " z = frac(x − μ, σ)$] #linebreak() #math.equation(block: false, alt: "x equals z σ plus μ")[$x = z σ + μ$]], [#strong[Central Limit Theorem] #math.equation(block: false, alt: "Z -score: z equals the fraction x bar minus μ over open parenthesis the fraction σ over the square root of n close parenthesis")[$Z "-score: " z = frac(overline(x) − μ, ( frac(σ, sqrt(n)) ))$]], [#strong[Normal Distribution Probabilities:]], [P(X ≤ x) = P(X \< x) Excel: =NORM.DIST(x,µ,σ,true) TI-84: normalcdf(-1E99,x,µ,σ)], [P(X ≥ x) = P(X \> x) Excel: = 1–NORM.DIST(x,µ,σ,true) TI-84: normalcdf(x,1E99,µ,σ)], [P(x#sub[1] ≤ X ≤ x#sub[2]) = P(x#sub[1]\< X \< x#sub[2]) = Excel: =NORM.DIST(x2,µ,σ,true)- NORM.DIST(x1,µ,σ,true) TI-84: normalcdf(x1,x2,µ,σ)], [#strong[Percentiles for Normal Distribution:]], [P(X ≤ x) = P(X \< x) Excel: =NORM.INV(area,µ,σ) TI-84: invNorm(area,µ,σ)], [P(X ≥ x) = P(X \> x) Excel: =NORM.INV(1–area,µ,σ) TI-84: invNorm(1–area,µ,σ)], [P(x#sub[1] ≤ X ≤ x#sub[2]) = P(x#sub[1] \< X \< x#sub[2]) = Excel: x#sub[1] =NORM.INV((1–area)/2,µ,σ) x#sub[2] =NORM.INV(1–((1–area)/2),µ,σ) TI-84: x#sub[1]= invNorm((1–area)/2,µ,σ) x#sub[2] =invNorm(1–((1–area)/2),µ,σ)], ))