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The Arcsine Transformation of a Binomial Random Variable

​
parameter n
10
parameter p
0.5
The arcsine transformation converts a binomial random variable into one that is nearly normal and whose variance depends very little on the parameter
p
. If
X
is a binomial random variable with parameters
n
and
p
(so
X
models the number of heads in
n
tosses of a coin that is weighted to have probability
p
of landing heads), then
Y=
-1
sin
X
n
is the arcsine transformation of
X
. The distribution of
Y
is nearly normal with mean
-1
sin
p
and variance
1/(4n)
. The histogram in the image is the distribution of
Y
with the rectangle heights scaled so that the sum of their areas is one, and the black curve is the approximating normal density function.
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