The uniform distribution is a probability distribution where all outcomes are equally likely. Use this calculator to find the probability between two values within a defined range, and also compute \(P(X \geq x)\) and \(P(X \leq x)\) for specific values.
P(x1 ≤ X ≤ x2):
P(X ≥ x1):
P(X ≤ x1):
P(X ≥ x2):
P(X ≤ x2):
Understanding the Uniform Distribution
The Uniform Distribution is a continuous probability distribution in which all outcomes between two limits are equally likely. It is often used to model scenarios where each outcome has the same chance of occurring within a specified range.
Key Components of the Uniform Distribution
- Lower Limit (a): The minimum value in the range of outcomes.
- Upper Limit (b): The maximum value in the range of outcomes.
- x1: The lower bound of the interval of interest within the uniform distribution.
- x2: The upper bound of the interval of interest within the uniform distribution.
Uniform Distribution Formula
The probability of a value falling between \( x_1 \) and \( x_2 \) in a uniform distribution is given by the formula:
Additionally, the cumulative distribution function (CDF) is used to calculate probabilities for \(P(X \geq x)\) and \(P(X \leq x)\):
- P(X ≤ x): \( \frac{x - a}{b - a} \) for \( a \leq x \leq b \)
- P(X ≥ x): \( \frac{b - x}{b - a} \) for \( a \leq x \leq b \)
Step-by-Step Example
Suppose we have a uniform distribution defined between \(a = 1\) and \(b = 10\), and we are interested in finding the probability that the value falls between \(x_1 = 2\) and \(x_2 = 5\). Additionally, we'll compute \(P(X \geq x_1)\), \(P(X \leq x_1)\), \(P(X \geq x_2)\), and \(P(X \leq x_2)\).
Step 1: Apply the Formula
Using the formula for \(P(x_1 \leq X \leq x_2)\):
Therefore, the probability that a value will fall between 2 and 5 is approximately 0.33333, or 33.33%.
Step 2: Compute Additional Probabilities
Using the CDF formulas for \(P(X \geq x)\) and \(P(X \leq x)\):
- P(X ≥ 2): \( \frac{10 - 2}{10 - 1} = \frac{8}{9} \approx 0.88889 \)
- P(X ≤ 2): \( \frac{2 - 1}{10 - 1} = \frac{1}{9} \approx 0.11111 \)
- P(X ≥ 5): \( \frac{10 - 5}{10 - 1} = \frac{5}{9} \approx 0.55556 \)
- P(X ≤ 5): \( \frac{5 - 1}{10 - 1} = \frac{4}{9} \approx 0.44444 \)
Further Reading
Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.