Discrete Uniform Distributions

Discrete Uniform Distributions#

A discrete uniform distribution is a probability distribution where all distinct outcomes within a finite range have equal probability of occurring, and outcomes outside this range have zero probability.

Discrete Uniform \((1, N)\)#

This form of the discrete uniform distribution has a lower bound of 1 and an upper bound of \(N\).

Probability Mass Function (PMF)#

The discrete uniform distribution has the following pmf:

\[\begin{split} p_X(x) = \begin{cases} \frac{1}{N} & \text{if } x = 1, 2, ..., N \\ 0 & \text{otherwise} \end{cases} \end{split}\]

Expectation#

From the pmf, we can derive \(E(X)\):

\[\begin{split} \begin{align} EX &= \sum_{x=1}^{N} \frac{1}{N} \ x \\ &= \frac{1}{N} \ \frac{N(N+1)}{2} \\ &= \frac{N+1}{2} \end{align} \end{split}\]

Variance#

We can also derive the variance \(Var(X)\):

\[ Var(X) = EX^2 - (EX)^2 \]

Since we’ve already found \(E(X)\), we only need to find \(E(X^2)\):

\[\begin{split} \begin{align} EX^2 &= \sum_{x=1}^N \frac{1}{N} \ x^2 \\ &= \frac{1}{N} \ \frac{N(N+1)(2N+1)}{6} \\ &= \frac{(N+1)(2N+1)}{6} \end{align} \end{split}\]

Now, we can solve for \(Var(X)\):

\[\begin{split} \begin{align} Var(X) &= EX^2 - (EX)^2 \\ &= \frac{(N+1)(2N+1)}{6} - \left[\frac{N+1}{2}\right]^2 \\ &= \frac{2N^2+3N+1}{6} - \frac{N^2+2N+1}{4} \\ &= \frac{N^2-1}{12} \\ \end{align} \end{split}\]

Discrete Uniform \((N_0, N_1)\)#

The discrete uniform distribution above can be generalized from \(\text{Uniform}(1, N)\) to \(\text{Uniform}(N_0, N_1)\).

\[\begin{split} X \sim \text{Uniform}(1, N_1 - N_0 + 1) \\ Y = X + (N_0 - 1)\sim \text{Uniform}(N_0, N_1) \end{split}\]

Probability Mass Function (PMF)#

\[\begin{split} p_X(x) = \begin{cases} \frac{1}{N_1 - N_0 + 1} & \text{if } x = N_0, N_0 + 1, ..., N_1 - 1, N_1 \\ 0 & \text{otherwise} \end{cases} \end{split}\]

Expectation#

We can use \(Y\) to derive the expectation of the distribution:

\[\begin{split} \begin{align} EY = E(X + N_0 - 1) &= EX + N_0 - 1 \\ &= \frac{N_1 - N_0 + 1 + 1}{2} + N_0 - 1 \\ &= \frac{N_1 + N_0}{2} \end{align} \end{split}\]

Variance#

Once again, we can use \(Y\) to derive the variance of the distribution:

\[\begin{split} \begin{align} Var(Y) = Var(X+N_0-1) &= Var(X) + N_0 - 1 \\ &= \frac{(N_1 - N_0 + 1)^2 - 1}{12} + N_0 - 1 \\ &= \frac{(N_1 - N_0)(N_1 - N_0 + 2)}{12} \end{align} \end{split}\]
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