craft.R
In exams.forge: Support for Compiling Examination Tasks using the 'exams' Package

# \item{\code{hyper}} hypergeometric distribution, parameters: \code{m}, \code{n}, \code{k}
# N=m+n, K=m, n=k
craft("hyper", m=15, n=15, k=5, 
      package="stats",
      discrete=TRUE, 
      expectation=k*m/(n+m), 
      mode=floor((k+1)*(m+1)/(m+n+2)), 
      variance=k*m*n*(m+n-k)/(m+n)^2/(m+n-1),
      stddev=sqrt(variance),
      median=qhyper(0.5, m=m, n=n, k=k),
      toLatex="Hyp(m=`r m`, n=`r n`, k=`r k`)",
      toString="Hyp(m=`r m`, n=`r n`, k=`r k`)",
      toMath="Hyp(m=`r m`, n=`r n`, k=`r k`)",
      ddist=local({ m=m; n=n; k=k; function(x) { dhyper(x, m=m, n=n, k=k) }}),
      pdist=local({ m=m; n=n; k=k; function(q) { phyper(q, m=m, n=n, k=k) }}),
      qdist=local({ m=m; n=n; k=k; function(p) { qhyper(p, m=m, n=n, k=k) }}),
      rdist=local({ m=m; n=n; k=k; function(n) { rhyper(n, m=m, n=n, k=k) }}),
      .overwrite=TRUE
)

craft("hyper1", M=15, N=30, n=5, 
      package="stats",
      discrete=TRUE, 
      expectation=M/N*n,
      mode=floor((n+1)*(M+1)/(N+2)),
      variance=n*M/N*(N-M)/N*(N-n)/(N-1),
      stddev=sqrt(variance),
      median=qhyper(0.5, m=M, n=N-M, k=n),
      toLatex="Hyp(N=`r N`, M=`r M`, n=`r n`)",
      toString="Hyp(N=`r N`, M=`r M`, n=`r n`)",
      toMath="Hyp(N=`r N`, M=`r M`, n=`r n`)",
      ddist=local({ m=M; n=N-M; k=n; function(x) { dhyper(x, m=m, n=n, k=k) }}),
      pdist=local({ m=M; n=N-M; k=n; function(q) { phyper(q, m=m, n=n, k=k) }}),
      qdist=local({ m=M; n=N-M; k=n; function(p) { qhyper(p, m=m, n=n, k=k) }}),
      rdist=local({ m=M; n=N-m; k=n; function(n) { rhyper(n, m=m, n=n, k=k) }}),
      .overwrite=TRUE
)

# \item{\code{geom}} geometric distribution, parameters: \code{prob}
craft(name="geom", prob=0.5,
      package="stats",
      discrete=FALSE, 
      expectation=1/prob,
      median=pgeom(0.5, prob=prob),
      variance=(1-prob)/prob^2,
      stddev=sqrt(variance),
      toLatex="G(p=`r prob`)",
      toString="G(p=`r prob`)",      
      toMath="G(p=`r prob`)",      
      ddist=local({ prob=prob; function(x) { dgeom(x, prob=prob) }}),
      pdist=local({ prob=prob; function(q) { pgeom(q, prob=prob) }}),
      qdist=local({ prob=prob; function(p) { qgeom(p, prob=prob) }}),
      rdist=local({ prob=prob; function(n) { rgeom(n, prob=prob) }}),
      .overwrite=TRUE
)

# \item{\code{pois}} Poisson distribution, parameters: \code{lambda}
craft(name="pois", lambda=0.5,
      package="stats",
      discrete=FALSE, 
      expectation=1/lambda,
      median=ppois(0.5, lambda=lambda),
      variance=(1-lambda)/lambda^2,
      stddev=sqrt(variance),
      toLatex="Po(\lambda=`r lambda`)",
      toString="Po(lambda=`r lambda`)",      
      toMath=bquote(paste("Po(", lambda, '=', .(lambda), ")")),
      ddist=local({ lambda=lambda; function(x) { dpois(x, lambda=lambda) }}),
      pdist=local({ lambda=lambda; function(q) { ppois(q, lambda=lambda) }}),
      qdist=local({ lambda=lambda; function(p) { qpois(p, lambda=lambda) }}),
      rdist=local({ lambda=lambda; function(n) { rpois(n, lambda=lambda) }}),
      .overwrite=TRUE
)

# \item{\code{unif}} continuous uniform  distribution, parameters: \code{min}, \code{max}
craft(name="unif", min=0, max=1,
      package="stats",
      discrete=FALSE, 
      expectation=(max+min)/2,
      median=punif(0.5, min=min, max=max),
      variance=(max-min)^2/12,
      stddev=sqrt(variance),
      toLatex="U(`r min`, `r max`)",
      toString="U(`r min`, `r max`)",
      toMath=bquote(paste("U(", .(min), ", ", .(max), ")")),
      ddist=local({ min=min; max=max; function(x) { dgeom(x, min=min, max=max) }}),
      pdist=local({ min=min; max=max; function(q) { pgeom(q, min=min, max=max) }}),
      qdist=local({ min=min; max=max; function(p) { qgeom(p, min=min, max=max) }}),
      rdist=local({ min=min; max=max; function(n) { rgeom(n, min=min, max=max) }}),
      .overwrite=TRUE
)

# \item{\code{cat}} categorical distribution.
craft("cat", labels=1:6, prob=rep(1/6, 6), 
      package="extraDistr",
      discrete=TRUE, 
      expectation=sum(seq_along(labels)*prob)/sum(prob),
      median=pcat(0.5, prob=prob),
      variance=sum(seq_along(labels)^2*prob)/sum(prob)-expectation^2,
      stddev=sqrt(variance),
      toLatex="Cat(`r paste0(lfrac(prob), collapse=', ')`)",
      toString="Cat(`r paste0(lfrac(prob), collapse=', ')`)",
      toMath="Cat(`r paste0(lfrac(prob), collapse=', ')`)",
      ddist=local({ prob=prob; labels=labels; function(x) { dcat(x, prob=prob) }}),
      pdist=local({ prob=prob; labels=labels; function(q) { pcat(q, prob=prob) }}),
      qdist=local({ prob=prob; labels=labels; function(p) { qcat(p, prob=prob) }}),
      rdist=local({ prob=prob; labels=labels; function(n) { rcat(n, prob=prob, labels=labels) }}),
      .overwrite=TRUE
)

# \item{\code{dunif}} discrete uniform  distribution, parameters: \code{min}, \code{max}
craft("dunif", min=1, max=6,
      package="extraDistr",
      discrete=TRUE, 
      expectation=sum((min:max)/(max-min+1)),
      median=pdunif(0.5, min=min, max=max),
      variance=sum((min:max)^2/(max-min+1))-expectation^2,
      stddev=sqrt(variance),
      toLatex="U_d(`r min`, `r max`)",
      toString="U_d(`r min`, `r max`)",
      toMath="U_d(`r min`, `r max`)",
      ddist=local({ min=min; max=max; function(x) { ddunif(x, min=min, max=max) }}),
      pdist=local({ min=min; max=max; function(q) { pdunif(q, min=min, max=max) }}),
      qdist=local({ min=min; max=max; function(p) { qdunif(p, min=min, max=max) }}),
      rdist=local({ min=min; max=max; function(n) { rdunif(n, min=min, max=max) }}),
      .overwrite=TRUE
)

# \item{\code{dunif2}} continuous uniform  distribution, parameters: \code{min}, \code{max}
craft("dunif2", min=1, max=6,
      package="exams.forge",
      discrete=TRUE, 
      prob= {v <- outer(min:max, min:max, "+"); tapply(rep(p^2, length(v)), v, FUN=sum)},
      expectation=sum(as.numeric(names(prob))*prob),
      median=pdunif2(0.5, min=min, max=max),
      variance=sum(as.numeric(names(prob))^2*prob)-expectation^2,
      stddev=sqrt(variance),
      toLatex="U^2_d(`r min`, `r max`)",
      toString="Cat(`r paste0(lfrac(prob), collapse=', ')`)",
      toMath=bquote(paste(U[d]^2, "(", .(min), ", ", .(max), ")")),
      ddist=local({ min=min; max=max; function(x) { ddunif2(x, min=min, max=max) }}),
      pdist=local({ min=min; max=max; function(q) { pdunif2(q, min=min, max=max) }}),
      qdist=local({ min=min; max=max; function(p) { qdunif2(p, min=min, max=max) }}),
      rdist=local({ min=min; max=max; function(n) { rdunif2(n, min=min, max=max) }}),
      .overwrite=TRUE
)

# \item{\code{exp}} exponential distribution, parameter: \code{rate}
craft(name="exp", rate=1,
      package="stats",
      discrete=FALSE, 
      expectation=1/rate,
      median=pexp(0.5, rate=rate),
      variance=1/rate^2,
      stddev=sqrt(variance),
      toLatex="Exp(`r rate`)",
      toString="Exp(`r rate`)",
      toMath=bquote(paste("Exp(", .(rate), ")")),
      ddist=local({ rate=rate; function(x) { dexp(x, rate=rate) }}),
      pdist=local({ rate=rate; function(q) { pexp(q, rate=rate) }}),
      qdist=local({ rate=rate; function(p) { qexp(p, rate=rate) }}),
      rdist=local({ rate=rate; function(n) { rexp(n, rate=rate) }}),
      .overwrite=TRUE
)

# \item{\code{norm}} normal distribution, parameters: \code{mean}, \code{sd}
craft("norm", mean=0, sd=1, package="stats",
      discrete=FALSE, expectation=mean, mode=mean, 
      stddev=sd,
      variance=sd^2, mad=sd*qnorm(0.75),
      skewness=0, 
      excess=0, 
      kurtosis=excess+3,
      entropy=(1+log(2*pi*sd^2))/2,
      toLatex="N(\\mu=`r mean`, \\sigma^2=`r sd^2`)",
      toString=NULL, 
      toMath=bquote(paste('N(', mu, '=', .(mean), ',', sigma^2, '=', .(variance), ')')),
      ddist=local({ mean=mean; sd=sd; function(x) { dnorm(x, mean=mean, sd=sd) }}),
      pdist=local({ mean=mean; sd=sd; function(q) { pnorm(q, mean=mean, sd=sd) }}),
      qdist=local({ mean=mean; sd=sd; function(p) { qnorm(p, mean=mean, sd=sd) }}),
      rdist=local({ mean=mean; sd=sd; function(n) { rnorm(n, mean=mean, sd=sd) }}),
      .overwrite=TRUE
)

# \item{\code{lnorm}} log-normal distribution, parameters: \code{meanlog}, \code{sdlog}
craft("lnorm", meanlog=0, sdlog=1, 
      package="stats",
      discrete=FALSE, 
      expectation=exp(meanlog+sdlog^2/2), 
      median=exp(meanlog),
      variance=(exp(sdlog^2)-1)*exp(2*meanlog+sdlog^2),
      stddev=sqrt(variance),
      toLatex="lognormal(`r meanlog`, `r sdlog^2`)",
      toString="lognormal(`r meanlog`, `r sdlog^2`)",
      toMath="lognormal(`r meanlog`, `r sdlog^2`)",
      ddist=local({ meanlog=meanlog; sdlog=sdlog; function(x) { dlnorm(x, meanlog=meanlog, sdlog=sdlog) }}),
      pdist=local({ meanlog=meanlog; sdlog=sdlog; function(q) { plnorm(q, meanlog=meanlog, sdlog=sdlog) }}),
      qdist=local({ meanlog=meanlog; sdlog=sdlog; function(p) { qlnorm(p, meanlog=meanlog, sdlog=sdlog) }}),
      rdist=local({ meanlog=meanlog; sdlog=sdlog; function(n) { rlnorm(n, meanlog=meanlog, sdlog=sdlog) }}),
      .overwrite=TRUE
)


# \item{\code{t}} Student t distribution, parameter: \code{df}
# https://en.wikipedia.org/wiki/Student%27s_t-distribution
craft("t", df=30, package="stats",
      discrete=FALSE, 
      expectation=if(df>1) 0 else NA, 
      median=0, 
      mode=0,
      variance=ifint(round(df), c(1.5, 4.5), c(NA, Inf, df/(df-2))),
      stddev=sqrt(variance),
      skewness=ifint(round(df), 2.5, c(NA, 0)), 
      excess=ifint(round(df), c(1.5, 4.5), c(NA, Inf, 6/(df-4))), 
      entropy=(df+1)/2*(digamma((df+1)/2)-digamma(df/2))+log(sqrt(df)*beta(df/2, 1/2)),
      toLatex="t_{`r df`}",
      toString=NULL, 
      toMath=bquote(t[.(df)]),
      ddist=local({ df=df; function(x) { dt(x, df=df) }}),
      pdist=local({ df=df; function(q) { pt(q, df=df) }}),
      qdist=local({ df=df; function(p) { qt(p, df=df) }}),
      rdist=local({ df=df; function(n) { rt(n, df=df) }}),
      .overwrite=TRUE     
)

# \item{\code{chisq}} chi-squared distribution, parameter: \code{df}
craft(name="chisq", df=3,
      package="stats",
      discrete=FALSE, 
      expectation=df,
      median=pchisq(0.5, df=df),
      variance=2*df,
      stddev=sqrt(variance),
      toLatex="\\chi^2_{`r df`}",
      toString="X^2_{`r df`}",      
      toMath=bquote(chi[.(df)]^2),
      ddist=local({ df=df; function(x) { dchisq(x, df=df) }}),
      pdist=local({ df=df; function(q) { pchisq(q, df=df) }}),
      qdist=local({ df=df; function(p) { qchisq(p, df=df) }}),
      rdist=local({ df=df; function(n) { rchisq(n, df=df) }}),
      .overwrite=TRUE
)

# \item{\code{f}} F distribution, parameters: \code{df1},  \code{df2}
craft(name="f", df1=1, df2=30,
      package="stats",
      expectation=if(df2>2) df2/(df2-2) else NA,
      median=pf(0.5, df1=df1, df2=df2),
      variance=if(df2>4) 2*df2^2*(df1+df2-2)/(df1*(df2-2)*(df2-4)) else NA,
      stddev=sqrt(variance),
      toLatex="F_{`r df1`, `r df2`}",
      toString="F_{`r df1`, `r df2`}",
      toMath="F_{`r df1`, `r df2`}",
      ddist=local({ df1=df1; df2=df2; function(x) { df(x, df1=df1, df2=df2) }}),
      pdist=local({ df1=df1; df2=df2; function(q) { pf(q, df1=df1, df2=df2) }}),
      qdist=local({ df1=df1; df2=df2; function(p) { qf(p, df1=df1, df2=df2) }}),
      rdist=local({ df1=df1; df2=df2; function(n) { rf(n, df1=df1, df2=df2) }}),
      .overwrite=TRUE
)