Bayesian Econometrics Software

B.11 Probability and cumulative density functions

The following two tables describe statements that are used to evaluate probability density/mass and cumulative density functions (pdfs and cdfs) of some popular distributions.


Syntax	Mathematical expression	Arguments and return values

W = betapdf(X, alpha, beta); W = betacdf(X, alpha, beta);	$f (x) = \frac{x^{α - 1} \cdot {(1 - x)}^{β - 1}}{B (α, β)}$ $F (x) = \frac{\int_{0}^{x} t^{α - 1} \cdot {(1 - t)}^{β - 1} d t}{B (α, β)}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the beta distribution with shape parameters alpha and beta, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with entries between zero and one If X contains negative entries or entries above one an error is produced alpha must be a positive number beta must be a positive number

W = chi2pdf(X, p); W = chi2cdf(X, p);	$f (x) = \frac{x^{\frac{p}{2} - 1} \cdot e^{- \frac{x}{2}}}{2^{\frac{p}{2}} \cdot Γ (\frac{p}{2})}$ $F (x) = \frac{\int_{0}^{x} t^{\frac{p}{2} - 1} \cdot e^{- \frac{t}{2}} d t}{2^{\frac{p}{2}} \cdot Γ (\frac{p}{2})}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the chi-squared distribution with p degrees of freedom, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with non-negative entries If X contains negative entries an error is produced p must be a positive number

W = exppdf(X, lambda); W = expcdf(X, lambda);	$f (x) = λ e^{- λ x}$ $F (x) = 1 - e^{- λ x}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the exponential distribution with rate parameter lambda, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with non-negative entries If X contains negative entries an error is produced lambda must be a positive number

W = evpdf(X, mu, sigma); W = evcdf(X, mu, sigma);	$f (x) = \frac{1}{σ} exp \{- z - e^{- z}\}$ $F (x) = exp \{- e^{- z}\}$ where: $z = \frac{x - μ}{σ}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the type-I extreme-value distribution with location parameter mu and scale parameter sigma, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset mu must be a number sigma must be a positive number

W = fpdf(X, p1, p2); W = fcdf(X, p1, p2);	$f (x) = \frac{{(\frac{{(p_{1} x)}^{p_{1}} p_{2}^{p_{2}}}{{(p_{1} x + p_{2})}^{p_{1} + p_{2}}})}^{\frac{1}{2}}}{x B (p_{1} 2, p_{2} 2)}$ $F (x) = \frac{\int_{0}^{\frac{p_{1} x}{p_{1} x + p_{2}}} t^{\frac{p_{1}}{2} - 1} {(1 - t)}^{\frac{p_{2}}{2} - 1} d t}{B (\frac{p_{1}}{2}, \frac{p_{1}}{2})}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of Fisher’s F distribution with numerator degrees of freedom p1 and denominator degrees of freedom p2, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with non-negative entries If X contains negative entries an error is produced p1 must be a positive number p2 must be a positive number

W = gampdf(X, alpha, beta); W = gamcdf(X, alpha, beta);	$f (x) = \frac{β^{α} \cdot x^{α - 1} \cdot e^{- β x}}{Γ (α)}$ $F (x) = \frac{β^{α} \int_{0}^{x} t^{α - 1} \cdot e^{- β t} d t}{Γ (α)}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the gamma distribution with shape parameter alpha and rate parameter beta, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with non-negative entries If X contains negative entries an error is produced alpha must be a positive number beta must be a positive number

W = logisticpdf(X, mu, s); W = logisticcdf(X, mu, s);	$f (x) = \frac{e^{- \frac{x - μ}{s}}}{s \cdot {(1 + e^{- \frac{x - μ}{s}})}^{2}}$ $F (x) = \frac{1}{1 + e^{- \frac{x - μ}{s}}}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the logistic distribution with mean mu and scale parameter s (variance equal to $\frac{s^{2} π^{2}}{3}$ ), evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset mu must be a number s must be a positive number

W = logitnpdf(X, mu, sigma); W = logitncdf(X, mu, sigma);	$f (x) = \frac{exp \{- \frac{{(log (\frac{x}{1 - x}) - μ)}^{2}}{2 σ^{2}}\}}{x (1 - x) \sqrt{2 π σ^{2}}}$ $F (x) = \int_{0}^{x} \frac{exp \{- \frac{{(log (\frac{t}{1 - t}) - μ)}^{2}}{2 σ^{2}}\}}{t (1 - t) \sqrt{2 π σ^{2}}} d t$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of logit-Normal distribution with location parameter mu and scale parameter sigma, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with entries between zero and one If X contains negative entries or entries greater than one an error is produced mu must be a number sigma must be a positive number

W = lognpdf(X, mu, sigma); W = logncdf(X, mu, sigma);	$f (x) = \frac{exp \{- \frac{{(log (x) - μ)}^{2}}{2 σ^{2}}\}}{x \sqrt{2 π σ^{2}}}$ $F (x) = \int_{0}^{x} \frac{exp \{- \frac{{(log (t) - μ)}^{2}}{2 σ^{2}}\}}{t \sqrt{2 π σ^{2}}} d t$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the log-Normal distribution with location parameter mu and scale parameter sigma, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with non-negative entries If X contains negative entries an error is produced mu must be a positive number sigma must be a positive number

W = normpdf(X, mu, sigma); W = normcdf(X, mu, sigma);	$f (x) = \frac{exp \{- \frac{{(x - μ)}^{2}}{σ^{2}}\}}{\sqrt{2 π σ^{2}}}$ $F (x) = \int_{- \infty}^{x} \frac{exp \{- \frac{{(t - μ)}^{2}}{σ^{2}}\}}{\sqrt{2 π σ^{2}}} d t$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the normal distribution with mean parameter mu and standard deviation sigma, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset mu must be number sigma must be a positive number

W = tpdf(X, p); W = tcdf(X, p);	$f (x) = \frac{Γ (\frac{p + 1}{2})}{\sqrt{p π} Γ (p 2)} {(1 + \frac{x^{2}}{p})}^{- \frac{p + 1}{2}}$ $F (x) = \int_{- \infty}^{x} \frac{Γ (\frac{p + 1}{2})}{\sqrt{p π} Γ (p 2)} {(1 + \frac{t^{2}}{p})}^{- \frac{p + 1}{2}} d t$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the t distribution with p degrees of freedom, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset p must be a positive number

W = truncnpdf(X, mu, sigma); W = truncncdf(X, mu, sigma);	$f (x) = \frac{\frac{1}{σ} ϕ (- \frac{μ}{σ})}{1 - Φ (- μ σ)}$ $F (x) = \frac{Φ (\frac{x - μ}{σ}) - Φ (- \frac{μ}{σ})}{1 - Φ (- μ σ)}$ where: $ϕ ()$ is the standard normal pdf $Φ ()$ is the standard normal cdf	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of a Normal distribution with location parameter mu and scale parameter sigma, truncated from below at zero and evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with non-negative entries mu must be a number sigma must be a positive number

W = wblpdf(X, alpha, beta); W = wblcdf(X, alpha, beta);	$f (x) = \frac{α}{β} {(\frac{x}{β})}^{α - 1} e^{- {(\frac{x}{β})}^{α}}$ $F (x) = 1 - e^{- {(\frac{x}{β})}^{α}}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the Weibull distribution with shape parameter alpha and scale parameter beta, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with non-negative entries If X contains negative entries an error is produced alpha must be a positive number beta must be a positive number


Syntax	Mathematical expression	Arguments and return values

W = binompdf(X, n, p); W = binomcdf(X, n, p);	$f (x) = \frac{n!}{x! (n - x)!} p^{x} {(1 - p)}^{n - x}$ $F (x) = \sum_{k = 0}^{x} \frac{n!}{k! (n - k)!} p^{k} {(1 - p)}^{n - k}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the binomial distribution with n of trials and probability of success in each trial p, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with integer entries between zero and n If X contains negative entries or entries above n an error is produced n must be a non-negative number p must be a number between zero and one Note that the function allows for non-integer values of n by replacing the factorials in the expression by the Gamma function.

W = nbinompdf(X, n, p); W = nbinomcdf(X, n, p);	$f (x) = \frac{(x + n - 1)!}{x! (n - 1)!} p^{x} {(1 - p)}^{n}$ $F (x) = \sum_{k = 0}^{x} \frac{(x + n - 1)!}{x! (n - 1)!} p^{x} {(1 - p)}^{n}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the negative-binomial distribution with n failures until stopping and probability of success in each trial p, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with non-negative integer entries If X contains negative entries an error is produced n must be a positive number p must be a number between zero and one Note that the function allows for non-integer values of n by replacing the factorials in the expression by the Gamma function.

W = poissonpdf(X, lambda); W = poissoncdf(X, lambda);	$f (x) = \frac{λ^{x} e^{- λ}}{x!}$ $F (x) = \frac{λ^{k} e^{- λ}}{k!}$	W is a matrix with dimensions equal to those of X and entries equal to the pdf/cdf of the Poisson distribution with rate parameter lambda, evaluated at each entry of X. The function works element-wise. X must be a matrix or dataset with non-negative integer entries If X contains negative entries an error is produced lambda must be a positive number

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