# Standard deviation, quartiles of Arcsine Distribution

I want to compute standard deviation,1st quartile, median and 3rd quartile of Arcsine distribution. I know its PDF is $g(x)=\frac{1}{\pi\sqrt{x(1-x)}},x\in [0,1]$

Its CDF is $\frac2\pi arcsin(\sqrt{x})$ and mean is $\frac12$ by symmetry.

Now, how shall I compute its standard deviation and quartiles? It will be good is someone can guide me.

• Where precisely did you get stuck? Commented Apr 9, 2016 at 16:44
• Another hint is that this distribution is a special case of the beta distribution, with both parameters equal to $\frac {1}{2}$ Commented Apr 10, 2016 at 14:11

The pdf is defined for $x \in [0,1]$ as

$$g(x) = \dfrac{1}{\pi \sqrt{x ( 1-x)}}.$$

You need to find the 1st quartile, median, 3rd quartile and the standard deviation. The median is the easiest, since you have already argued that the mean is 1/2 since the distribution is symmetric around 1/2. Use a similar argument to find the median.

The quartiles can be found by using the definition. Let $q_1$ and $q_3$ be the 1st and 3rd quartile. Then by definition $q_1$ and $q_3$ are such, so that

$$\int_0^{q_1}g(x)\, dx = .25 \quad \text{ and } \quad \int_{0}^{q_3}g(x) \, dx = .75$$

These integrals are a little complicated, but you do not need to solve them, since you have the CDF available. Hint: What is the definition of CDF?

The standard deviation is the square root of the variance, and the variance can be calculated by first finding the second moment

$$\int_0^1 x^2 g(x) \, dx.$$

Everything should follow straight from the definitions.