Calculate Uniform Distribution

Uniform Distribution Calculator

Use this calculator to determine the mean, variance, standard deviation, and probabilities for a continuous uniform distribution defined by its minimum and maximum values.

Probability Calculations

Enter values below to calculate specific probabilities. These are optional.

function calculateUniformDistribution() { var minValue = parseFloat(document.getElementById('minValue').value); var maxValue = parseFloat(document.getElementById('maxValue').value); var specificValue = document.getElementById('specificValue').value; var lowerBound = document.getElementById('lowerBound').value; var upperBound = document.getElementById('upperBound').value; var resultsDiv = document.getElementById('uniformDistributionResult'); resultsDiv.innerHTML = "; // Clear previous results if (isNaN(minValue) || isNaN(maxValue)) { resultsDiv.innerHTML = 'Please enter valid numbers for Minimum and Maximum Values.'; return; } if (minValue >= maxValue) { resultsDiv.innerHTML = 'Minimum Value must be less than Maximum Value.'; return; } var range = maxValue – minValue; // Calculate Mean var mean = (minValue + maxValue) / 2; // Calculate Variance var variance = Math.pow(range, 2) / 12; // Calculate Standard Deviation var stdDev = Math.sqrt(variance); // Calculate Probability Density Function (PDF) value var pdfValue = 1 / range; var outputHTML = '

Results:

'; outputHTML += 'Mean (E[X]): ' + mean.toFixed(4) + "; outputHTML += 'Variance (Var[X]): ' + variance.toFixed(4) + "; outputHTML += 'Standard Deviation (σ): ' + stdDev.toFixed(4) + "; outputHTML += 'Probability Density Function (f(x)): ' + pdfValue.toFixed(4) + ' (for ' + minValue.toFixed(2) + ' ≤ x ≤ ' + maxValue.toFixed(2) + ')'; // Calculate P(X <= x) if (specificValue !== '') { var x = parseFloat(specificValue); if (isNaN(x)) { outputHTML += 'Specific Value (x) is not a valid number. Skipping P(X ≤ x) calculation.'; } else { var probX_le_x; if (x maxValue) { probX_le_x = 1; } else { probX_le_x = (x – minValue) / range; } outputHTML += 'P(X ≤ ' + x.toFixed(2) + '): ' + probX_le_x.toFixed(4) + "; } } // Calculate P(x1 <= X x2) { outputHTML += 'Lower Bound (x1) must be less than or equal to Upper Bound (x2) for P(x1 ≤ X ≤ x2).'; } else { var prob_x1_le_X_le_x2; var effectiveX1 = Math.max(minValue, x1); var effectiveX2 = Math.min(maxValue, x2); if (effectiveX1 >= effectiveX2) { // No overlap or x1 is after x2 within the range prob_x1_le_X_le_x2 = 0; } else { prob_x1_le_X_le_x2 = (effectiveX2 – effectiveX1) / range; } outputHTML += 'P(' + x1.toFixed(2) + ' ≤ X ≤ ' + x2.toFixed(2) + '): ' + prob_x1_le_X_le_x2.toFixed(4) + "; } } resultsDiv.innerHTML = outputHTML; } .uniform-distribution-calculator { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; box-shadow: 0 2px 4px rgba(0,0,0,0.1); } .uniform-distribution-calculator h2 { color: #333; text-align: center; margin-bottom: 20px; } .uniform-distribution-calculator h3 { color: #555; margin-top: 25px; margin-bottom: 15px; border-bottom: 1px solid #eee; padding-bottom: 5px; } .uniform-distribution-calculator label { display: inline-block; margin-bottom: 5px; font-weight: bold; width: 250px; /* Align labels */ } .uniform-distribution-calculator input[type="number"] { width: calc(100% – 260px); /* Adjust width considering label */ padding: 8px; margin-bottom: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .uniform-distribution-calculator button { background-color: #007bff; color: white; padding: 10px 15px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; display: block; width: 100%; margin-top: 20px; } .uniform-distribution-calculator button:hover { background-color: #0056b3; } .calculator-results { margin-top: 20px; padding: 15px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 5px; } .calculator-results p { margin: 8px 0; line-height: 1.5; color: #333; } .calculator-results p strong { color: #0056b3; }

Understanding the Uniform Distribution

The uniform distribution is a type of probability distribution where all outcomes are equally likely within a defined interval. It's often referred to as a rectangular distribution because its probability density function (PDF) forms a rectangle when plotted. This distribution is characterized by two parameters: a minimum value (a) and a maximum value (b).

Key Characteristics and Formulas

For a continuous uniform distribution over the interval [a, b]:

• Probability Density Function (PDF):

  The PDF, denoted as f(x), describes the likelihood of a random variable taking on a given value. For a uniform distribution, the density is constant across the interval.

  f(x) = 1 / (b - a) for a ≤ x ≤ b

  f(x) = 0 otherwise

• Mean (Expected Value):

  The mean, or expected value (E[X]), represents the average value of the distribution. For a uniform distribution, it's simply the midpoint of the interval.

  E[X] = (a + b) / 2

• Variance:

  Variance (Var[X]) measures the spread of the distribution. A higher variance indicates that the data points are more spread out from the mean.

  Var[X] = (b - a)^2 / 12

• Standard Deviation:

  The standard deviation (σ) is the square root of the variance and provides a more interpretable measure of spread in the same units as the data.

  σ = √((b - a)^2 / 12)

• Cumulative Distribution Function (CDF):

  The CDF, denoted as F(x), gives the probability that a random variable X will take a value less than or equal to x, i.e., P(X ≤ x).

  • If x < a, then F(x) = 0
  • If a ≤ x ≤ b, then F(x) = (x - a) / (b - a)
  • If x > b, then F(x) = 1
• Probability of an Interval:

  The probability that X falls within a specific sub-interval [x1, x2] (where a ≤ x1 ≤ x2 ≤ b) is calculated as:

  P(x1 ≤ X ≤ x2) = (x2 - x1) / (b - a)

When is Uniform Distribution Used?

The uniform distribution is often used in situations where:

• There is no prior knowledge about the likelihood of outcomes within a given range, so all outcomes are assumed to be equally probable.
• Modeling random numbers, such as those generated by a computer.
• Approximating other distributions when only the range is known.
• In simulations, to generate random variables for various processes.

Example Scenario

Imagine a bus arrives at a stop every 10 minutes, and you arrive at the stop at a random time. The waiting time for the bus can be modeled by a uniform distribution. If the bus arrives at 0, 10, 20 minutes past the hour, your waiting time could be anywhere between 0 and 10 minutes, with each waiting time being equally likely.

• Minimum Value (a): 0 minutes
• Maximum Value (b): 10 minutes

Using the calculator with these values:

• Mean Waiting Time: (0 + 10) / 2 = 5 minutes
• Variance: (10 – 0)^2 / 12 = 100 / 12 ≈ 8.33
• Standard Deviation: √8.33 ≈ 2.89 minutes
• Probability Density: 1 / (10 – 0) = 0.1

If you wanted to know the probability of waiting less than 3 minutes (P(X ≤ 3)):

• P(X ≤ 3): (3 – 0) / (10 – 0) = 3 / 10 = 0.3 (or 30%)

If you wanted to know the probability of waiting between 2 and 7 minutes (P(2 ≤ X ≤ 7)):

• P(2 ≤ X ≤ 7): (7 – 2) / (10 – 0) = 5 / 10 = 0.5 (or 50%)

This calculator helps you quickly compute these essential statistics and probabilities for any given uniform distribution.