How to calculate the median and the quantiles of this distribution?

Question

A median of a distribution of one random variable $X$ of the discrete or continuous type is a value of $x$ such that $P(X < x) \leq 1/2$ and $P(X \leq x) \geq 1/2$. If there is only one such $x$, it is called the median of the distribution.

Similarly, for $0 < p < 1$, a $(100p)$ th percentile (quantile of order p) of the distribution of the random variable $X$ is a value $\xi_p$ such that $P(X < \xi_p) \leq p$ and $P(X \leq \xi_p) \geq p$.

How to find these quantities for the random variable $X$ with probability mass function (pmf) given by $$ p(x) \colon = \frac{4!}{x! (4-x)!} \left(\frac{1}{4}\right)^x \left(\frac{3}{4}\right)^{4-x} $$ for $x = 0, 1, 2, 3, 4$.

Answer 1

This a binomial distribution with $n=4$, $p=\frac14$, and so mean $1$.

The probabilities and cumulative probabilities are

x   p(x)=P(X=x)  P(X <= x) 
0   0.31640625   0.31640625
1   0.42187500   0.73828125
2   0.21093750   0.94921875
3   0.04687500   0.99609375
4   0.00390625   1.00000000

so it should be easy to see that the median is $1$, the first quartile ($25$th percentile) is $0$ and the third quartile ($75$th percentile) is $2$. The other quantiles are similarly easy, looking at the cumulative probabilities.