*Last modified: September 21, 2024*

*This article is written in: ðŸ‡ºðŸ‡¸*

## Uniform Distribution

A continuous random variable X follows a uniform distribution over an interval [a, b] if it has a constant probability density over that interval. The uniform distribution is denoted as $X \sim \text{Uniform}(a, b)$.

### Probability Density Function (PDF)

The PDF of a uniform distribution is given by:

$$f(x) = \begin{cases} \frac{1}{b-a}, & \text{if}\ a \le x \le b \\ 0, & \text{otherwise} \end{cases} $$

### Cumulative Distribution Function (CDF)

The CDF of a uniform distribution is given by:

$$F(x) = \begin{cases} 0, & \text{if}\ x < a \\ \frac{x-a}{b-a}, & \text{if}\ a \le x \le b \\ 1, & \text{if}\ x > b \end{cases} $$

### Expected Value and Variance

The expected value (mean) of a uniform distribution is the midpoint of the interval [a, b]:

$$E[X] = \frac{a+b}{2}$$

The variance of a uniform distribution is given by:

$$\text{Var}(X) = \frac{(b-a)^2}{12}$$

### Moment Generating Functions and Moments

The moment generating function (MGF) of a uniform distribution is:

$$M_X(t) = E[e^{tX}] = \frac{e^{tb} - e^{ta}}{t(b-a)}$$

To find the n-th moment, we take the n-th derivative of the MGF with respect to t and then evaluate it at t=0:

$$E[X^n] = \frac{d^n M_X(t)}{dt^n}\Bigg|_{t=0}$$

**First Moment (Mean):**

$$E[X] = \frac{a+b}{2}$$

**Second Moment (Variance + Mean^2):**

$$E[X^2] = \frac{a^2 + ab + b^2}{3}$$

### Example: Timing in Simulation Game

In a simulation game, a developer wants to introduce an event occurring randomly between 2 and 4 seconds after a certain action. This timing follows a continuous uniform distribution.

Given parameters:

- Lower bound $a = 2$ seconds
- Upper bound $b = 4$ seconds

I. Probability of Event Occurring Within 2.5 Seconds:

For a continuous uniform distribution $U(a, b)$, the probability density function is $f(x; a, b) = \frac{1}{b - a}$ for $a \leq x \leq b$.

To find the probability of the event happening before 2.5 seconds:

$$ P(X < 2.5) = \frac{2.5 - 2}{4 - 2} = 0.25 $$

This means there is a 25% chance the event will occur within 2.5 seconds.

II. Expected Value and Variance of the Event Timing:

The expected value $E[X]$ and variance $\text{Var}(X)$ for a continuous uniform distribution are calculated as:

- Expected value (mean time):

$$ E[X] = \frac{a + b}{2} = \frac{2 + 4}{2} = 3 \text{ seconds} $$

- Variance:

$$ \text{Var}(X) = \frac{(b - a)^2}{12} = \frac{(4 - 2)^2}{12} = \frac{1}{3} \text{ seconds}^2 $$

These calculations help the developer understand the timing dynamics of the event in the simulation game.

### Applications

Uniform distributions are often used as a simple model when all possible outcomes are equally likely, such as in random number generators or when sampling from a finite set of values.