# The Arcsine Transformation of a Binomial Random Variable

The Arcsine Transformation of a Binomial Random Variable

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

p

X

n

p

X

n

p

Y=

-1

sin

X

n

X

Y

-1

sin

p

1/(4n)

Y