R 中的密度直方图

Question

I'm new to R and probability&statistics. I have a question regarding the histograms...

hist(rbinom(10000, 10, 0.1), freq=FALSE)

it shows the histogram following diagram which is not clear to me:

if the y-axis is density, so the total number should be %100, am I wrong?
But in the histogram, I can see that it is bigger than %100.

Answer 1

The area under the curve should be 1. Since your boxes appear to have width 1/2, the sum of the heights should be 2. To make this make more sense, use the breaks parameter to hist

hist(rbinom(10000, 10, 0.1), freq=FALSE, breaks = 5)

Or maybe even better

hist(rbinom(10000, 10, 0.1), freq=FALSE, breaks=seq(-0.5,5.5,1))

Answer 2

You can integrate the density function estimated based on your sample. The answer is approximately 1, so no contradiction.

set.seed(444)

s <- rbinom(10000, 10, 0.1)
dens_s <- table(s)/sum(table(s))
sum(as.numeric(names(dens_s))*dens_s)

Answer 3

Function hist returns a list object with all information necessary to answer the question.

I will set the RNG seed to make the example reproducible.

set.seed(1234)
h <- hist(rbinom(10000, 10, 0.1), freq=FALSE)

str(h)
#List of 6
# $ breaks  : num [1:11] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ...
# $ counts  : int [1:10] 3448 3930 0 1910 0 588 0 112 0 12
# $ density : num [1:10] 0.69 0.786 0 0.382 0 ...
# $ mids    : num [1:10] 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 4.25 4.75
# $ xname   : chr "rbinom(10000, 10, 0.1)"
# $ equidist: logi TRUE
# - attr(*, "class")= chr "histogram"

The relevant list members are breaks and density .

1. breaks is a vector of length 11, so there are 10 bins.
2. density is a vector of length 10, each corresponding to one of the bins.

Now compute the area of each bar by multiplying the bins lengths by the respective densities.

diff(h$breaks)    # bins lengths
# [1] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
h$density*diff(h$breaks)
# [1] 0.3448 0.3930 0.0000 0.1910 0.0000 0.0588 0.0000 0.0112 0.0000 0.0012

Total area:

sum(h$density*diff(h$breaks))
#[1] 1