📘 Chapter 4 Review — PDFs, CDFs & Excel

Normal \( \mathcal{N}(\mu,\sigma^2) \)

PDF: \[ f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \]
CDF: \(F(x)=\Phi\!\left(\dfrac{x-\mu}{\sigma}\right)\) (no closed form; use tables/software)
Mean/Var: \( \mu,\ \sigma^2 \)
Excel:
=NORM.DIST(x, mean, sd, TRUE) CDF
=NORM.DIST(x, mean, sd, FALSE) PDF
=NORM.S.DIST(z, TRUE) / =NORM.S.DIST(z, FALSE)
=NORM.INV(p, mean, sd), =NORM.S.INV(p)

Lognormal \( \mathrm{Lognormal}(\theta,\omega) \) where \(\ln X \sim \mathcal{N}(\theta,\omega^2)\)

PDF: \[ f(x)=\frac{1}{x\,\omega\sqrt{2\pi}}\exp\!\left(-\frac{(\ln x-\theta)^2}{2\omega^2}\right),\quad x>0 \]
CDF: \(F(x)=\Phi\!\left(\dfrac{\ln x-\theta}{\omega}\right)\)
Mean/Var: \(E[X]=e^{\theta+\omega^2/2},\;\; \mathrm{Var}(X)=e^{2\theta+\omega^2}(e^{\omega^2}-1)\)
Quantile: \(x_p=\exp\!\big(\theta+\omega\,\Phi^{-1}(p)\big)\)
Excel:
=LOGNORM.DIST(x, theta, omega, TRUE/FALSE)
=LOGNORM.INV(p, theta, omega)

Exponential (rate \(\lambda\)) or (mean \(\delta\))

Rate form \(X\sim \mathrm{Exp}(\lambda)\): PDF \(f(x)=\lambda e^{-\lambda x}\), CDF \(F(x)=1-e^{-\lambda x}\), \(E[X]=1/\lambda\).
Scale form \(X\sim \mathrm{Exp}(\delta)\): PDF \(f(x)=\frac{1}{\delta}e^{-x/\delta}\), CDF \(F(x)=1-e^{-x/\delta}\), \(E[X]=\delta\).
Excel:
=EXPON.DIST(x, lambda, TRUE/FALSE) λ = rate
(If you have mean δ, use lambda = 1/δ)

Gamma / Erlang

Shape–Scale \(X\sim \mathrm{Gamma}(\alpha,\theta)\): PDF \(f(x)=\dfrac{x^{\alpha-1}e^{-x/\theta}}{\Gamma(\alpha)\theta^\alpha}\), \(E[X]=\alpha\theta\), \(\mathrm{Var}(X)=\alpha\theta^2\).

Shape–Rate \(X\sim \mathrm{Gamma}(\alpha,\lambda)\): PDF \(f(x)=\dfrac{\lambda^\alpha x^{\alpha-1}e^{-\lambda x}}{\Gamma(\alpha)}\), \(E[X]=\alpha/\lambda\), \(\mathrm{Var}(X)=\alpha/\lambda^2\).

Erlang: integer \(\alpha=r\), waiting time to r-th Poisson event.
Excel (scale θ = “beta” in Excel):
=GAMMA.DIST(x, alpha, theta, TRUE/FALSE)
=GAMMA.INV(p, alpha, theta)
If you have rate λ, set theta = 1/λ.

Weibull \( \mathrm{Weibull}(\beta,\delta) \)

PDF: \[ f(x)=\frac{\beta}{\delta}\left(\frac{x}{\delta}\right)^{\beta-1} \exp\!\left(-\left(\frac{x}{\delta}\right)^\beta\right),\quad x>0 \]
CDF: \(F(x)=1-\exp\!\left(-\left(\frac{x}{\delta}\right)^\beta\right)\)
Mean/Var: \(E[X]=\delta\,\Gamma(1+1/\beta)\), \(\mathrm{Var}(X)=\delta^2\left[\Gamma(1+2/\beta)-\Gamma(1+1/\beta)^2\right]\)
Quantile: \(x_p=\delta[-\ln(1-p)]^{1/\beta}\)
Excel:
=WEIBULL.DIST(x, alpha, beta, TRUE/FALSE) alpha=shape=β, beta=scale=δ
=WEIBULL.INV(p, alpha, beta)

Beta \( \mathrm{Beta}(\alpha,\beta) \) and Scaled Beta \([a,b]\)

Standard Beta (0–1): \[ f(x)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)},\quad E[X]=\frac{\alpha}{\alpha+\beta},\ \ \mathrm{Var}(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} \]
Scaled Beta \([a,b]\): \(Y=a+(b-a)X\). \[ E[Y]=a+(b-a)\frac{\alpha}{\alpha+\beta},\quad \mathrm{Var}(Y)=(b-a)^2\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} \]
Excel:
=BETA.DIST(x, alpha, beta, TRUE/FALSE, [A], [B])
=BETA.INV(p, alpha, beta, [A], [B])

Chi-Square \( \chi^2_k \)

PDF: \[ f(x)=\frac{1}{2^{k/2}\Gamma(k/2)}x^{k/2-1}e^{-x/2},\quad x>0 \]
CDF: \(F(x)=\gamma(k/2,x/2)/\Gamma(k/2)\) (regularized)
Mean/Var: \(E[X]=k,\ \mathrm{Var}(X)=2k\)
Excel:
=CHISQ.DIST(x, df, TRUE/FALSE)
=CHISQ.DIST.RT(x, df) right tail
=CHISQ.INV(p, df), =CHISQ.INV.RT(p, df)

🧾 Excel Summary (All Distributions)

Normal

=NORM.DIST(x, mean, sd, TRUE/FALSE)
=NORM.S.DIST(z, TRUE/FALSE)
=NORM.INV(p, mean, sd), =NORM.S.INV(p)

Lognormal

=LOGNORM.DIST(x, theta, omega, TRUE/FALSE)
=LOGNORM.INV(p, theta, omega)

Exponential

=EXPON.DIST(x, lambda, TRUE/FALSE) (lambda = 1/mean)

Gamma / Erlang

=GAMMA.DIST(x, alpha, theta, TRUE/FALSE) (theta = scale)
=GAMMA.INV(p, alpha, theta)

Weibull

=WEIBULL.DIST(x, alpha, beta, TRUE/FALSE) (alpha=shape β, beta=scale δ)
=WEIBULL.INV(p, alpha, beta)

Beta

=BETA.DIST(x, alpha, beta, TRUE/FALSE, [A], [B])
=BETA.INV(p, alpha, beta, [A], [B])

Chi-Square

=CHISQ.DIST(x, df, TRUE/FALSE)
=CHISQ.DIST.RT(x, df) (right tail)
=CHISQ.INV(p, df), =CHISQ.INV.RT(p, df)

Parameter reminders: Excel’s “beta” for Gamma/Weibull means **scale**. For Gamma with rate \(\lambda\), use theta = 1/lambda. For Exponential with mean \(\delta\), use lambda = 1/δ. For Beta on \([A,B]\), pass [A],[B]; omit for standard \([0,1]\).