stochastically greater

Definition

A cdf $F_X$ is stochastically greater than a cdf $F_Y$ if $F_X(t) \le F_Y(t)$ for all $t$ and $F_X(t) <F_Y(t) $ for some $t$.

That is, $X$ tends to be bigger than $Y$. Now, we investigate the relationship between discrete random variables and uniform random variables.

Corollary

Let $X$ be a discrete random variable with cdf $F_X(x)$ and define the random variables $Y$ as $Y=F_X(X)$ then, $Y$ is stochastically greatly than a $uniform(0,1)$.

Proof

We assume that $X$ have the distribution that

$$
\begin{equation}

X \sim \left(

\begin{aligned}

x_1 & & x_2 & \dots & x_n\

p_1 & & p_2 & \dots & p_n

\end{aligned}

\right)

\end{equation}
$$

then we can get $F_X$

$$
\begin{equation}

F_X(t)= \left{

\begin{aligned}

&0 & & t<x_1\

&p_1 & & x_1\le t<x_2\

&\vdots & & \vdots \

&p_1+p_2+\dots+p_{n-1} & & x_{n-1}\le t<x_n\

&p_1+p_2+\dots+p_{n}=1 & & t\ge x_n

\end{aligned}

\right.

\end{equation}
$$

$Y=F_X(X)$ so, $P(Y>y)=P(F_X(X)>y)=P(X:F_X(X)>y)$

For example
$$
\begin{aligned}
P(Y>p_1+p_2+\dots+p_{n-1}+\frac{1}{2}p_n)=& P(F_X(X)>p_1+p_2+\dots+p_{n-1}+\frac{1}{2}p_n)\
= & P(X:F_X(X)>p_1+p_2+\dots+p_{n-1}+\frac{1}{2}p_n)\
= &P(X=a_n)=p_n
\end{aligned}
$$

Like this, we can prove Y is stochastically greater than uniform distribution. This Corollary can be used in proofing the Theorem in p-value.