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(x _{1} ≤ X ≤ x_{2}):**

**P(X ≥ x _{1}):**

**P(X ≤ x _{1}):**

**P(X ≥ x _{2}):**

**P(X ≤ x _{2}):**

## 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.**x**The lower bound of the interval of interest within the uniform distribution._{1}:**x**The upper bound of the interval of interest within the uniform distribution._{2}:

### 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.