Calculates density of the binomial distribution where support for 0 has been removed, and which we refer to as the (zero-) truncated binomial distribution (e.g., Rider, 1955; Stephan, 1945).
Arguments
- x
vector of values
- size
(scalar) number of trials (
n); must be 1 or more- prob
(scalar) probability of success on each trial (
p); must be nonzero
Details
Let \(X\) denote a random variable with probability mass function (pmf) of
\(\mathrm{p}(k;n,p) = \mathrm{Pr}(X = k; n, p) = \frac{\binom{n}{k} p^k (1-p)^{n-k}}{1 - (1-p)^n} \propto \binom{n}{k} p^k (1-p)^{n-k}\),
for \(k = 1, 2, ..., n\), and with zero mass, otherwise. We say that \(X\) is truncated binomial with parameters \((n,p)\), written as \(X \sim TBinom(n,p)\), and where \(n\) refers to the number of trials and \(p\) refers to the probability of success of each trial for the corresponding binomial distribution.
References
Rider, P. R. (1955). Truncated binomial and negative binomial distributions. Journal of the American Statistical Association, 50(271), 877-883.
Stephan, F. F. (1945). The expected value and variance of the reciprocal and other negative powers of a positive Bernoullian variate. The Annals of Mathematical Statistics, 16(1), 50-61.
See also
calc_moments_truncbinom() for computing TBinom moments
Examples
dtruncbinom(1:20, 20, .2)
#> [1] 5.831844e-02 1.385063e-01 2.077594e-01 2.207444e-01 1.765955e-01
#> [6] 1.103722e-01 5.518610e-02 2.241935e-02 7.473118e-03 2.055107e-03
#> [11] 4.670699e-04 8.757560e-05 1.347317e-05 1.684146e-06 1.684146e-07
#> [16] 1.315739e-08 7.739642e-10 3.224851e-11 8.486450e-13 1.060806e-14
#same as above
dbinom(1:20, size = 20, prob = .2) / sum(dbinom(1:20, size = 20, prob = .2))
#> [1] 5.831844e-02 1.385063e-01 2.077594e-01 2.207444e-01 1.765955e-01
#> [6] 1.103722e-01 5.518610e-02 2.241935e-02 7.473118e-03 2.055107e-03
#> [11] 4.670699e-04 8.757560e-05 1.347317e-05 1.684146e-06 1.684146e-07
#> [16] 1.315739e-08 7.739642e-10 3.224851e-11 8.486450e-13 1.060806e-14