1. What is the Uniform Probability Density Function?

The uniform probability density function describes a continuous probability distribution where all outcomes between a lower bound (a) and upper bound (b) are equally likely. It's fundamental in probability theory and statistics.

2. How Does the Calculator Work?

The calculator uses the uniform probability density function:

Where:

• \( a \) — Lower bound of the distribution
• \( b \) — Upper bound of the distribution
• \( f(x) \) — Probability density at any point x between a and b

Explanation: The function is constant between a and b, and zero elsewhere. The area under the curve always equals 1.

3. Importance of Uniform Distribution

Details: The uniform distribution is used in simulations, random sampling, and as a null model in statistical tests. It's the simplest continuous probability distribution.

4. Using the Calculator

Tips: Enter the lower bound (a) and upper bound (b) values. The upper bound must be greater than the lower bound.

5. Frequently Asked Questions (FAQ)

Q1: What does the uniform probability density represent?
A: It represents a scenario where all values within the interval [a, b] are equally likely to occur.

Q2: What's the difference between uniform and normal distribution?
A: Uniform has constant probability between bounds, while normal is bell-shaped with higher probability near the mean.

Q3: When is the uniform distribution used in real applications?
A: In simulations, random number generation, and when modeling complete uncertainty about which outcome will occur.

Q4: What happens if a equals b?
A: The function becomes undefined (division by zero) as there's no interval width.

Q5: How is this related to the uniform cumulative distribution function?
A: The CDF is the integral of the PDF, giving the probability that a random variable is less than or equal to a certain value.