# Class "Beta" The Beta distribution with parameters `shape1` $= a$ and `shape2` $= b$ has density [REMOVE_ME]$$ f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}%Gamma(a+b)/(Gamma(a)Gamma(b))x^(a-1)(1-x)^(b-1) [REMOVE_ME_2]$$ for $a > 0$, $b > 0$ and $0 <= x <= 1$ where the boundary values at $x=0$ or $x=1$ are defined as by continuity (as limits). class ## Description The Beta distribution with parameters `shape1` $= a$ and `shape2` $= b$ has density $$ f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}%Gamma(a+b)/(Gamma(a)Gamma(b))x^(a-1)(1-x)^(b-1) $$ for $a > 0$, $b > 0$ and $0 <= x <= 1$ where the boundary values at $x=0$ or $x=1$ are defined as by continuity (as limits). ## Note The non-central Beta distribution is defined (Johnson et al, 1995, pp. 502) as the distribution of $X/(X+Y)$ where $X ~ chi^2_2a(lambda)$ and $Y ~ chi^2_2b$. C.f. `rbeta` ## Ad hoc methods For R Version `<2.3.0` ad hoc methods are provided for slots `q`, `r` if `ncp!=0`; for R Version `>=2.3.0` the methods from package `stats` are used. ## Objects from the Class Objects can be created by calls of the form `Beta(shape1, shape2)`. This object is a beta distribution. ## Slots - **`img`**: Object of class `"Reals"`: The space of the image of this distribution has got dimension 1 and the name "Real Space". - **`param`**: Object of class `"BetaParameter"`: the parameter of this distribution (shape1 and shape2), declared at its instantiation - **`r`**: Object of class `"function"`: generates random numbers (calls function rbeta) - **`d`**: Object of class `"function"`: density function (calls function dbeta) - **`p`**: Object of class `"function"`: cumulative function (calls function pbeta) - **`q`**: Object of class `"function"`: inverse of the cumulative function (calls function qbeta) - **`.withArith`**: logical: used internally to issue warnings as to interpretation of arithmetics - **`.withSim`**: logical: used internally to issue warnings as to accuracy - **`.logExact`**: logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function - **`.lowerExact`**: logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function - **`Symmetry`**: object of class `"DistributionSymmetry"`; used internally to avoid unnecessary calculations. ## Extends Class `"AbscontDistribution"`, directly. Class `"UnivariateDistribution"`, by class `"AbscontDistribution"`. Class `"Distribution"`, by class `"AbscontDistribution"`. ## Methods - **initialize**: `signature(.Object = "Beta")`: initialize method - **shape1**: `signature(object = "Beta")`: returns the slot shape1 of the parameter of the distribution - **shape1<-**: `signature(object = "Beta")`: modifies the slot shape1 of the parameter of the distribution - **shape2**: `signature(object = "Beta")`: returns the slot shape2 of the parameter of the distribution - **shape2<-**: `signature(object = "Beta")`: modifies the slot shape2 of the parameter of the distribution - **`-`**: `signature(e1 = "numeric", e2 = "Beta")` if `ncp(e2)==0` and `e1 == 1`, an exact (central) `Beta(shape1 = shape2(e2), shape2 = shape1(e2))` is returned, else the default method is used; exact ## Author(s) Thomas Stabla statho3@web.de , Florian Camphausen fcampi@gmx.de , Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de , Matthias Kohl Matthias.Kohl@stamats.de ## See Also `BetaParameter-class` `AbscontDistribution-class` `Reals-class` `rbeta` ## Examples ```r B <- Beta(shape1 = 1, shape2 = 1) # B is a beta distribution with shape1 = 1 and shape2 = 1. r(B)(1) # one random number generated from this distribution, e.g. 0.6979795 d(B)(1) # Density of this distribution is 1 for x=1. p(B)(1) # Probability that x < 1 is 1. q(B)(.1) # Probability that x < 0.1 is 0.1. shape1(B) # shape1 of this distribution is 1. shape1(B) <- 2 # shape1 of this distribution is now 2. Bn <- Beta(shape1 = 1, shape2 = 3, ncp = 5) # Bn is a beta distribution with shape1 = 1 and shape2 = 3 and ncp = 5. B0 <- Bn; ncp(B0) <- 0; # B0 is just the same beta distribution as Bn but with ncp = 0 q(B0)(0.1) ## q(Bn)(0.1) ## => from R 2.3.0 on ncp no longer ignored... ## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.) ```

Beta-class function