Computation of probabilities (Binomial and hypergeometric distribution) using R

We will compute probabilities of Binomial probability distribution

Usage

dbinom(x, size, prob, log = FALSE)
pbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)
qbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE)
rbinom(n, size, prob)

Arguments

column 1	column 2
x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
size	number of trials (zero or more).
prob	probability of success on each trial.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X \le x]P[X≤x], otherwise, P[X > x]P[X>x].

Details

The binomial distribution with size = n=n and prob = p=p has density
p(x) = {n \choose x} {p}^{x} {(1-p)}^{n-x}p(x)=(
x
n
)p^x(1−p)^n−x

for x = 0, \ldots, nx=0,…,n. Note that binomial coefficients can be computed by choose in R.

If an element of x is not integer, the result of dbinom is zero, with a warning.

p(x)p(x) is computed using Loader's algorithm, see the reference below.

The quantile is defined as the smallest value xx such that F(x) \ge pF(x)≥p, where FF is the distribution function.

Values

dbinom gives the density, pbinom gives the distribution function, qbinom gives the quantile function and rbinom generates random deviates.

If size is not an integer, NaN is returned.

The length of the result is determined by n for rbinom, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Example problems

A) Let X- B(n= 10,p = 0.3). Find P(2 < X < 6) and P(X=>5)

n = 10;n
p = 0.3;p
x = 3:5;x # P[2<X<6] = p[x=3] + p[x=4] + p[x=5]
ProbA.1 = sum(dbinom(x,n,p));ProbA.1
x = 5:10;x # To find P[x => 5]
ProbA.2 = sum(dbinom(x,n,p));ProbA.2
# ProbA.2 = 1 - pbinom(4,n,p);Prob1.2 . P[X <= x] = pbinom(x, n, p).

B) Let X-B(n = 5,p = 0.2). Find P(X= 4) and P(X <= 2)

n = 5;n
p = 0.2;p
x = 4;x # p[X = 4]
ProbB.1 = dbinom(x,n,p);ProbB.1
ProbB.2 = pbinom(2,n,p); # P[X <= 2]

C) Let X - B(n=5, p=0.2). Find k such that P(X=> k) = 0.2552

n = 5;n
p = 0.2;p
k=qbinom(0.2552,n,p);k

Hypergeometric distribution

Usage

dhyper(x, m, n, k, log = FALSE)
phyper(q, m, n, k, lower.tail = TRUE, log.p = FALSE)
qhyper(p, m, n, k, lower.tail = TRUE, log.p = FALSE)
rhyper(nn, m, n, k)

Arguments

Details

The hypergeometric distribution is used for sampling without replacement. The density of this distribution with parameters m, n and k (named NpNp, N-NpN−Np, and nn, respectively in the reference below, where N := m+nN:=m+n is also used in other references) is given by

p(x)=(
x
m
)(
k−x
n
)/(
k
m+n
)

for x = 0, \ldots, kx=0,…,k.

Note that p(x)p(x) is non-zero only for \max(0, k-n) \le x \le \min(k, m)max(0,k−n)≤x≤min(k,m)

Example problems

D) Let X - Hyp(N = 12,M = 8, n=3). Find P(X < 2) and P(X=2).

N = 12;N
M = 8;M
n = 3;n
x = 0:1;x # P[X < 2]
ProbD.1 = sum(dhyper(x, N - M, M, n));ProbD.1
x = 2; # P[X = 2]
ProbD.2 = dhyper(x,N-M,M,n);ProbD.2

E) Let X - Hyp(N = 20, M = 9, n = 5). Find P(3 <= X <= 6) and P(X => 4).

N = 20;N
M = 9;M
n = 5;n
x = 3:6;x # P[3 <= X <= 6]
ProbE.1 = sum(dhyper(x,N-M,M,n));ProbE.1
ProbE.2 = 1 - phyper(3, N-M,M,n);ProbE.2

Computation of probabilities (Binomial and hypergeometric distribution) using R — Software.

We will compute probabilities of Binomial probability distribution

Details

Values

Example problems

Hypergeometric distribution

Details

Example problems