A Uniform distribution over an interval $(a, b)$, also called the continuous uniform distribution, is a probability distribution in which all values within the interval $[a, b]$ are equally likely to occur. Mathematically, it is denoted as:

$X \sim \mathcal{U}(a, b)$

Probability Density Function (PDF):

The probability density function of a uniform distribution over $(a, b)$ is defined as:

\begin{cases} \frac{1}{b - a} & \text{for } a \leq x \leq b, \\ 0 & \text{otherwise.} \end{cases}$$ This means that the probability of the random variable $$X$$ taking any value within the interval $$[a, b]$$ is constant and equal to $$\frac{1}{b - a}$$. ### **Cumulative Distribution Function (CDF):** The cumulative distribution function, $$F(x)$$, of a uniform distribution is the probability that the random variable $$X$$ will take a value less than or equal to a given value $$x$$: $$F(x) = \begin{cases} 0 & \text{for } x < a, \\ \frac{x - a}{b - a} & \text{for } a \leq x \leq b, \\ 1 & \text{for } x > b. \end{cases}$$ This function grows linearly from 0 to 1 as $$x$$ moves from $$a$$ to $$b$$. ### **Procedure for Drawing a Random Sample:** To draw a random sample from a uniform distribution over $$(a, b)$$, follow this procedure: 1. **Generate a random number $$U$$**: First, generate a random number $$U$$ from a standard uniform distribution $$\mathcal{U}(0, 1)$$, which produces numbers in the interval $$[0, 1]$$. 2. **Transform the variable**: Use the following transformation to map $$U$$ into the interval $$(a, b)$$: $$X = a + (b - a) \cdot U$$ The variable $$X$$ will now follow the uniform distribution $$\mathcal{U}(a, b)$$, where $$a \leq X \leq b$$. This procedure leverages the fact that any uniform distribution can be generated by scaling and shifting a standard uniform distribution. ### Summary: - **PDF**: $$f(x) = \frac{1}{b - a}$$ for $$a \leq x \leq b$$. - **CDF**: $$F(x) = \frac{x - a}{b - a}$$ for $$a \leq x \leq b$$. - **Random sample generation**: Use the transformation $$X = a + (b - a) \cdot U$$ where $$U$$ is uniformly distributed over $$[0, 1]$$. Would you like more details or have any further questions? --- Here are 5 related questions you might find interesting: 1. How does the mean and variance of the uniform distribution over $$(a, b)$$ compare to other distributions? 2. What is the expected value and variance of a uniform distribution $$\mathcal{U}(a, b)$$? 3. How is the uniform distribution related to other distributions, such as the normal distribution? 4. Can you describe the discrete uniform distribution and its differences from the continuous uniform distribution? 5. How can the uniform distribution be used in simulation or Monte Carlo methods? **Tip**: In computational tasks, uniform distribution is frequently used as a basis for generating other, more complex distributions through transformations.