Last modified: February 17, 2018
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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.