Is there any "classical" distribution that is considered bimodal? For example, "Normal" is unimodal, "Gamma" is unimodal.
If I have to generate a sample of 100 numbers from a univariate bimodal distribution, how should I proceed with that? I think that "sticking" two samples from unimodal distributions doesn't really work, or am I wrong?
Will a sample created from sticking two random samples be considered independent?
$\endgroup$1 Answer
$\begingroup$First, beta distributions with both shape parameters below 1 are bimodal. The support of a beta distribution is $(0,1),$ and these beta distributions have probability concentrated near $0$ and $1$.
Second, mixtures of normal distributions can be bimodal, roughly speaking, if the two normal distributions being mixed have means that are several standard deviations apart. For example, a 50:50 mixture of $\mathsf{Norm}(\mu=5, \sigma=2)$ and$\mathsf{Norm}(\mu=10, \sigma=1)$ is noticeably bimodal. (The mixture distribution is has a density function that is the average of the density functions of the two distributions being mixed. If the two SDs are equal, you may want to investigate the exact separation between the means for bimodality.)
I'm not sure what you mean by 'sticking' samples together. Certainly, if you just concatenate a sample of size 50 from $\mathsf{Unif}(0,1)$ with a sample of size 50 from $\mathsf{Unif}(2,4),$ you will get a bimodal histogram with 100 observations, but I personally wouldn't consider that to be random sampling from a bimodal distribution. If you randomly select 100 times from one of two different random samples (each of size 100) at each step, then the result is a random sample from a mixture distribution, which may be bimodal.
Here is R code to get samples of size $n = 500$ from a beta distribution and a bimodal normal mixture distribution, along with histograms of the two datasets, with the bivariate densities superimposed.
Notes: (1) I use $n = 500$ instead of $n = 100$ just for
illustration, so you can see that the histograms are close to matching the
bimodal densities. (2) This is very simplistic R code, which you should adapt
to whatever software you're using and to your degree of sophistication using it.
(3) I don't know if you are allowed to use pre-programed functions, such asrbeta and rnorm, that sample from specific distributions, or whether
you are supposed to use the inverse-CDF method with uniform observations.
set.seed(1234) # include seed for same samples n = 500; x1 = rbeta(n, .5, .4) y1 = rnorm(n, 5, 2); y2 = rnorm(n, 10, 1) w = rbinom(n, 1, .5) # 50:50 random choice x2 = w*y1 + (1-w)*y2 # normal mixture par(mfrow=c(1,2)) # two panels per plot hist(x1, prob=T, col="skyblue2", main="BETA(.5, .4)"); rug(x1) curve(dbeta(x, .5, .4), lwd=2, col="blue", add=T) hist(x2, prob=T, col="skyblue2", main="Normal Mixture"); rug(x2) curve(.5*dnorm(x,5,2)+.5*dnorm(x,10,1), lwd=2, col="blue", add=T) par(mfrow=c(1,1)) # return to default single panelAddendum: (After Comments) It seems you define a bimodal PDF to mean two achieved maximums of equal height. Slight modifications in the code are as follows, with additional comments at the changes:
set.seed(1235)
n = 2000; u1 = rbeta(n, .5, .5) # 2000 wasteful, but simple
cond = u1 > .01 & u1 < .99
v1 = u1[cond]; x1 = v1[1:500] # truncate and use 500
n = 500; y1 = rnorm(n, 2, 2); y2 = rnorm(n, 12, 2) # same SD
w = rbinom(n, 1, .5)
x2 = w*y1 + (1-w)*y2
par(mfrow=c(1,2)) hist(x1, prob=T, col="skyblue2", main="Truncated BETA(.5, .5)"); rug(x1) k = diff(pbeta(c(.01,.99),.5,.5)) # adj for truncation curve((1/k)*dbeta(x, .5, .5),.01,.99, lwd=2, col="blue", add=T) hist(x2, prob=T, col="skyblue2",ylim=c(0,.1), main="Normal Mixture"); rug(x2) curve(.5*dnorm(x,2,2)+.5*dnorm(x,12,2), lwd=2, col="blue", add=T)
par(mfrow=c(1,1)) 
