📘 Chapter 4: Continuous Random Variables and Probability Distributions

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💬 Excel

  • Chapter 4 Excel Functions

    • 📢 Student Q&A

    Frequently asked questions about Chapter 4 topics.

    Q0a: How do I write the PDF of a normal distribution, and what do μ and σ mean? Student asked

    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.

    Excel tip: PDF at x: =NORM.DIST(x, μ, σ, FALSE) • CDF: =NORM.DIST(x, μ, σ, TRUE)

    Q0b: In Excel’s DIST functions, what happens if I use 1 (TRUE) vs 0 (FALSE)? Student asked

    A: The last argument is the cumulative flag.

    Examples:

    Reminder: in finance functions (PV, PMT, etc.), the last argument is type: 0=end (ordinary annuity), 1=beginning (annuity due)—not a cumulative flag.

    Q0c: Normal probabilities — P(X < a), P(X > b), and P(a < X < b) Student asked

    Model: If X ~ N(μ, σ²), standardize with z = (x−μ)/σ and use the standard normal CDF Φ.

    Continuous note: P(X ≤ c) = P(X < c) for normal. Using ≤ in Excel is fine.

    Quick example (μ=100, σ=15):

    Uniform • Exponential • Lognormal • Normal

    Uniform distribution

    Q1: How do I write the model for a uniform random variable?

    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.

    Q2: How do I compute a uniform probability P(X ≤ x)?

    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.

    Q3: What is the mean and variance of a uniform distribution?

    A: Mean = (a + b)/2, Variance = (b − a)² / 12. These are always true for continuous uniform.

    Q4: How do I find the PDF value at a point for uniform?

    A: For any x in [a,b], f(x) = 1/(b−a). Outside [a,b], f(x)=0.

    Q5: How do I set up Excel for a uniform probability?

    A: Use =(x−a)/(b−a) for cumulative probability. Excel has no built-in UNIF.DIST for continuous uniform, so use this simple formula.

    Exponential distribution

    Q6: What is the definition of an exponential distribution?

    A: X is exponential if it has PDF f(x)=λe^{−λx}, x≥0. Mean = 1/λ. Write X ~ Exp(λ).

    Q7: How do I compute P(X > t) for exponential?

    A: Use the survival function: P(X>t)=e^{−λt}. Example: if λ=0.2, P(X>3)=e^{−0.6}=0.5488.

    Q8: How do I compute P(X ≤ t) for exponential?

    A: Use CDF: F(t)=1−e^{−λt}. In Excel: =EXPON.DIST(t, λ, TRUE).

    Q9: How do I find a percentile for exponential?

    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.

    Q10: How do I get λ from mean?

    A: λ = 1/mean. Example: mean=4 ⇒ λ=0.25.

    Lognormal distribution

    Q11: What is a lognormal distribution?

    A: X is lognormal if ln(X) is normal: ln X ~ N(μ,σ²). Model notation: ln X ~ N(μ, σ²).

    Q12: How do I compute P(X ≤ x) for lognormal?

    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).

    Q13: What is the median of a lognormal?

    A: Median = e^{μ}. This is always true because median(ln X)=μ.

    Q14: How do I find a lognormal percentile?

    A: Use x_p = exp( μ + σ z_p ). Example: for p=0.9, z_p=1.2816 ⇒ x_p = exp(μ+σz_p).

    Q15: How do I do lognormal calculations in Excel?

    A: Use LOGNORM.DIST(x, μ, σ, TRUE) for CDF, and =EXP(μ+σ*NORM.S.INV(p)) for percentiles.

    Normal distribution

    Q16: How do I write the model for a normal variable?

    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).

    Q17: How do I convert a value x to a z-score?

    A: z = (x − μ) / σ. This standardizes x so you can use the standard normal table Φ(z).

    Q18: How do I find P(X ≤ x) for a normal distribution?

    A: Convert x to z and look up Φ(z). In Excel: =NORM.DIST(x, μ, σ, TRUE).

    Q19: How do I find P(X > x)?

    A: P(X > x) = 1 − P(X ≤ x). Compute the CDF first, then subtract from 1.

    Q20: How do I find P(a ≤ X ≤ b)?

    A: P(a ≤ X ≤ b) = Φ((b−μ)/σ) − Φ((a−μ)/σ). Shade between a and b.

    Q21: How do I find a percentile xₚ for normal?

    A: xₚ = μ + σ zₚ, where zₚ = Φ⁻¹(p). In Excel: =NORM.INV(p, μ, σ).

    Q22: What is the 95% central interval for a normal variable?

    A: Cut 2.5% in each tail: [μ − 1.96σ, μ + 1.96σ].

    Q23: How do I find zₐ for a given tail area?

    A: Use zₐ = Φ⁻¹(1−α). Example: upper 5% tail ⇒ z₀․₉₅ = 1.645. In Excel: =NORM.S.INV(0.95).

    Q24: How do I check if my answer makes sense?

    A: Draw a normal curve, mark μ, shade the area you are finding, and confirm probability is reasonable (left tail small, middle big, etc.).

    Q25: What is the probability within ±1σ of μ?

    A: P(|X − μ| ≤ σ) ≈ 0.6827 (68.3%). This is the “68–95–99.7 rule.”

    Q26: What is the probability within ±2σ?

    A: ≈ 0.9545 (95.45%).

    Q27: What is the probability within ±3σ?

    A: ≈ 0.9973 (99.73%). Very rare to be outside ±3σ.

    Q28: How do I find μ or σ if given a probability?

    A: Set up equation: P(X ≤ x) = p ⇒ (x − μ)/σ = zₚ ⇒ solve for μ or σ depending on which is unknown.

    Q29: How do I do standard normal probabilities in Excel?

    A: Use =NORM.S.DIST(z, TRUE) for CDF and =1-NORM.S.DIST(z,TRUE) for upper tail.

    Q30: How do I invert standard normal probabilities?

    A: Use =NORM.S.INV(p) in Excel to get z for lower-tail probability p.