1. What is Continuous Uniform Distribution?

The Continuous Uniform Distribution is a probability distribution where all intervals of the same length have equal probability. It's characterized by constant probability density between a lower bound (a) and upper bound (b).

2. How Does the Calculator Work?

The calculator uses the uniform distribution formula:

Where:

• \( a \) — Lower bound of the distribution
• \( b \) — Upper bound of the the distribution
• \( x \) — Value at which to evaluate the probability density

Explanation: The probability density is constant within the interval [a, b] and zero outside this interval.

3. Applications of Uniform Distribution

Details: Uniform distribution is used in various fields including simulation modeling, random number generation, and quality control where events are equally likely to occur within a specific range.

4. Using the Calculator

Tips: Enter the lower bound (a), upper bound (b), and the x value at which you want to evaluate the probability density. Ensure that b > a for valid calculations.

5. Frequently Asked Questions (FAQ)

Q1: What is the mean of a uniform distribution?
A: The mean is \( \frac{a + b}{2} \), which is the midpoint of the interval.

Q2: What is the variance of a uniform distribution?
A: The variance is \( \frac{(b - a)^2}{12} \).

Q3: Can the uniform distribution be used for discrete values?
A: Yes, but that would be the discrete uniform distribution where each value has equal probability.

Q4: What are some real-world examples of uniform distribution?
A: Random number generation, arrival times in queuing theory, and measurement errors with equal likelihood across a range.

Q5: How is the cumulative distribution function defined?
A: The CDF is \( F(x) = \frac{x - a}{b - a} \) for \( a \leq x \leq b \).