Download Chapter 4 Slides (ppt_chapter4.pptx)
Each quiz gives instant feedback and auto-scoring. Click a quiz to open it in a new tab.
Download the Word files to write answers, then compare with solutions for self-study.
Frequently asked questions about Chapter 4 topics.
A: Model notation: X ~ N(μ, σ²)
.
The probability density function (PDF) is:
f(x) = (1 / (σ√(2π))) · exp(−(x − μ)² / (2σ²))
, for all real x.
σ = √(σ²)
(take the square root of the variance).Excel tip: PDF at x: =NORM.DIST(x, μ, σ, FALSE)
• CDF: =NORM.DIST(x, μ, σ, TRUE)
A: The last argument is the cumulative flag.
TRUE
or 1
⇒ return the CDF P(X ≤ x)
(area to the left).FALSE
or 0
⇒ return the PDF height f(x)
at that x (a density, not a probability by itself).Examples:
=NORM.DIST(x, μ, σ, TRUE)
→ CDF; =NORM.DIST(x, μ, σ, FALSE)
→ PDF.=NORM.S.DIST(z, TRUE)
→ Φ(z); =NORM.S.DIST(z, FALSE)
→ ϕ(z).=LOGNORM.DIST(x, μ, σ, TRUE)
→ CDF; =LOGNORM.DIST(x, μ, σ, FALSE)
→ PDF.=EXPON.DIST(x, λ, TRUE)
→ 1−e^(−λx)
; =EXPON.DIST(x, λ, FALSE)
→ λe^(−λx)
.
Reminder: in finance functions (PV
, PMT
, etc.), the last argument is type
:
0
=end (ordinary annuity), 1
=beginning (annuity due)—not a cumulative flag.
Model: If X ~ N(μ, σ²), standardize with z = (x−μ)/σ and use the standard normal CDF Φ.
=NORM.DIST(a, μ, σ, TRUE)
or =NORM.S.DIST((a-μ)/σ, TRUE)
=1 - NORM.DIST(b, μ, σ, TRUE)
or =1 - NORM.S.DIST((b-μ)/σ, TRUE)
=NORM.DIST(b, μ, σ, TRUE) - NORM.DIST(a, μ, σ, TRUE)
Continuous note: P(X ≤ c) = P(X < c) for normal. Using ≤ in Excel is fine.
Quick example (μ=100, σ=15):
=NORM.DIST(120,100,15,TRUE)
≈ 0.9082=1 - NORM.DIST(85,100,15,TRUE)
≈ 0.8413=NORM.DIST(110,100,15,TRUE) - NORM.DIST(90,100,15,TRUE)
≈ 0.4950Uniform • Exponential • Lognormal • Normal
A: Write X ~ Unif[a,b], where a is the minimum and b is the maximum. The PDF is f(x)=1/(b−a) for a ≤ x ≤ b.
A: Use the CDF F(x)=(x−a)/(b−a). Example: if X ~ Unif[20,40], P(X ≤ 30) = (30−20)/(40−20) = 0.5.
A: Mean = (a + b)/2, Variance = (b − a)² / 12. These are always true for continuous uniform.
A: For any x in [a,b], f(x) = 1/(b−a). Outside [a,b], f(x)=0.
A: Use =(x−a)/(b−a) for cumulative probability. Excel has no built-in UNIF.DIST for continuous uniform, so use this simple formula.
A: X is exponential if it has PDF f(x)=λe^{−λx}, x≥0. Mean = 1/λ. Write X ~ Exp(λ).
A: Use the survival function: P(X>t)=e^{−λt}. Example: if λ=0.2, P(X>3)=e^{−0.6}=0.5488.
A: Use CDF: F(t)=1−e^{−λt}. In Excel: =EXPON.DIST(t, λ, TRUE).
A: Solve x_p = −ln(1−p)/λ. Example: for p=0.9, λ=0.25, x=−ln(0.1)/0.25=9.21 min.
A: λ = 1/mean. Example: mean=4 ⇒ λ=0.25.
A: X is lognormal if ln(X) is normal: ln X ~ N(μ,σ²). Model notation: ln X ~ N(μ, σ²).
A: Convert to ln X: P(X ≤ x) = P(ln X ≤ ln x) = Φ((ln x − μ)/σ). In Excel: =LOGNORM.DIST(x, μ, σ, TRUE).
A: Median = e^{μ}. This is always true because median(ln X)=μ.
A: Use x_p = exp( μ + σ z_p ). Example: for p=0.9, z_p=1.2816 ⇒ x_p = exp(μ+σz_p).
A: Use LOGNORM.DIST(x, μ, σ, TRUE) for CDF, and =EXP(μ+σ*NORM.S.INV(p)) for percentiles.
A: Write X ~ N(μ, σ²), where μ is the mean and σ² is the variance. Example: test scores with mean 80 and sd 10 ⇒ X ~ N(80, 10²).
A: z = (x − μ) / σ. This standardizes x so you can use the standard normal table Φ(z).
A: Convert x to z and look up Φ(z). In Excel: =NORM.DIST(x, μ, σ, TRUE).
A: P(X > x) = 1 − P(X ≤ x). Compute the CDF first, then subtract from 1.
A: P(a ≤ X ≤ b) = Φ((b−μ)/σ) − Φ((a−μ)/σ). Shade between a and b.
A: xₚ = μ + σ zₚ, where zₚ = Φ⁻¹(p). In Excel: =NORM.INV(p, μ, σ).
A: Cut 2.5% in each tail: [μ − 1.96σ, μ + 1.96σ].
A: Use zₐ = Φ⁻¹(1−α). Example: upper 5% tail ⇒ z₀․₉₅ = 1.645. In Excel: =NORM.S.INV(0.95).
A: Draw a normal curve, mark μ, shade the area you are finding, and confirm probability is reasonable (left tail small, middle big, etc.).
A: P(|X − μ| ≤ σ) ≈ 0.6827 (68.3%). This is the “68–95–99.7 rule.”
A: ≈ 0.9545 (95.45%).
A: ≈ 0.9973 (99.73%). Very rare to be outside ±3σ.
A: Set up equation: P(X ≤ x) = p ⇒ (x − μ)/σ = zₚ ⇒ solve for μ or σ depending on which is unknown.
A: Use =NORM.S.DIST(z, TRUE) for CDF and =1-NORM.S.DIST(z,TRUE) for upper tail.
A: Use =NORM.S.INV(p) in Excel to get z for lower-tail probability p.