Continuous Uniform Distribution Calculator
Probability Calculations
Calculation Results:
'; outputHTML += 'Lower Bound (a): ' + lowerBoundA.toFixed(4) + "; outputHTML += 'Upper Bound (b): ' + upperBoundB.toFixed(4) + "; outputHTML += 'Mean (E[X]): ' + mean.toFixed(4) + "; outputHTML += 'Variance (Var[X]): ' + variance.toFixed(4) + "; outputHTML += 'Standard Deviation (SD[X]): ' + stdDev.toFixed(4) + "; outputHTML += 'Probability Density Function (f(x) for a ≤ x ≤ b): ' + pdfValue.toFixed(4) + "; // Probability Results section outputHTML += 'Probability Results:
'; // P(X <= x) calculation if (!isNaN(xValue)) { if (xValue < lowerBoundA) { outputHTML += 'P(X ≤ ' + xValue.toFixed(4) + '): 0.0000 (x is below the lower bound)'; } else if (xValue > upperBoundB) { outputHTML += 'P(X ≤ ' + xValue.toFixed(4) + '): 1.0000 (x is above the upper bound)'; } else { var cdfValue = (xValue – lowerBoundA) / range; outputHTML += 'P(X ≤ ' + xValue.toFixed(4) + '): ' + cdfValue.toFixed(4) + "; } } else { outputHTML += 'P(X ≤ x): Please enter a valid number for x.'; } // P(c <= X = dValue) { outputHTML += 'For P(c ≤ X ≤ d), c must be less than d.'; } else { // Adjust c and d to be within [a, b] for calculation var effectiveC = Math.max(lowerBoundA, cValue); var effectiveD = Math.min(upperBoundB, dValue); if (effectiveC >= effectiveD) { // This means the interval [c,d] does not overlap with [a,b] or is invalid outputHTML += 'P(' + cValue.toFixed(4) + ' ≤ X ≤ ' + dValue.toFixed(4) + '): 0.0000 (Interval [' + cValue.toFixed(4) + ', ' + dValue.toFixed(4) + '] does not overlap with [' + lowerBoundA.toFixed(4) + ', ' + upperBoundB.toFixed(4) + '])'; } else { var probabilityInterval = (effectiveD – effectiveC) / range; outputHTML += 'P(' + cValue.toFixed(4) + ' ≤ X ≤ ' + dValue.toFixed(4) + '): ' + probabilityInterval.toFixed(4) + "; } } } else { outputHTML += 'P(c ≤ X ≤ d): Please enter valid numbers for c and d.'; } resultsDiv.innerHTML = outputHTML; }Understanding the Continuous Uniform Distribution
The continuous uniform distribution, often referred to as the rectangular distribution, is a fundamental concept in probability theory and statistics. It describes a scenario where all outcomes within a given interval are equally likely. Unlike discrete distributions where outcomes are distinct points, the continuous uniform distribution applies to a continuous range of values.
What is a Continuous Uniform Distribution?
Imagine a random number generator that produces a number between 0 and 10. If every number in that range (e.g., 0.1, 5.7, 9.999) has an equal chance of being selected, then the numbers follow a continuous uniform distribution. The probability density function (PDF) for a continuous uniform distribution over an interval [a, b] is constant across the interval and zero elsewhere.
The PDF is given by:
f(x) = 1 / (b - a) for a ≤ x ≤ b
f(x) = 0 otherwise
Here, 'a' is the lower bound and 'b' is the upper bound of the interval. The total area under the PDF curve is always 1, representing the total probability.
Key Characteristics and Formulas
This calculator helps you determine the key statistical measures and probabilities associated with a continuous uniform distribution:
- Mean (Expected Value, E[X]): This is the average value you would expect from the distribution. For a uniform distribution, it's simply the midpoint of the interval.
E[X] = (a + b) / 2 - Variance (Var[X]): This measures the spread of the distribution around its mean. A larger variance indicates a wider spread of possible outcomes.
Var[X] = (b - a)² / 12 - Standard Deviation (SD[X]): The square root of the variance, providing a measure of spread in the same units as the data.
SD[X] = √((b - a)² / 12) - Probability Density Function (PDF, f(x)): As mentioned, this is the constant height of the distribution within the interval [a, b].
f(x) = 1 / (b - a) - Cumulative Distribution Function (CDF, P(X ≤ x)): This gives the probability that a random variable X will take a value less than or equal to a specific value 'x'.
P(X ≤ x) = (x - a) / (b - a)fora ≤ x ≤ bP(X ≤ x) = 0forx < aP(X ≤ x) = 1forx > b - Probability of an Interval (P(c ≤ X ≤ d)): This calculates the probability that a random variable X falls within a specific sub-interval [c, d] within the main interval [a, b].
P(c ≤ X ≤ d) = (d - c) / (b - a)fora ≤ c ≤ d ≤ b
How to Use the Calculator
Simply input the lower bound (a) and upper bound (b) of your continuous uniform distribution. You can also specify a value 'x' to find the cumulative probability P(X ≤ x), or a range [c, d] to find the probability P(c ≤ X ≤ d). The calculator will then provide the mean, variance, standard deviation, PDF value, and the requested probabilities.
Example Scenario
Consider a bus that 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 as a continuous uniform distribution between 0 and 10 minutes.
- Lower Bound (a): 0 minutes
- Upper Bound (b): 10 minutes
Using the calculator with these values:
- Mean Waiting Time: (0 + 10) / 2 = 5 minutes
- Variance: (10 – 0)² / 12 = 100 / 12 ≈ 8.3333
- Standard Deviation: √8.3333 ≈ 2.8868 minutes
- PDF (f(x)): 1 / (10 – 0) = 0.1000
Now, let's say you want to find the probability that you wait less than or equal to 3 minutes (P(X ≤ 3)):
- Value for P(X ≤ x) (x): 3
- P(X ≤ 3): (3 – 0) / (10 – 0) = 3 / 10 = 0.3000
What is the probability that you wait between 2 and 7 minutes (P(2 ≤ X ≤ 7))?
- Lower bound for P(c ≤ X ≤ d) (c): 2
- Upper bound for P(c ≤ X ≤ d) (d): 7
- P(2 ≤ X ≤ 7): (7 – 2) / (10 – 0) = 5 / 10 = 0.5000
This calculator provides a quick and accurate way to explore the properties and probabilities of continuous uniform distributions for various applications in engineering, science, and business.