Visual Tools
Calculators
Tables
Mathematical Keyboard
Converters
Other Tools

Continuous Distributions Calculator


Probability Calculators
Discrete Distributions Calculator
Statistics Calculator

Continuous Uniform Distribution Calculator

Calculate probabilities and distribution properties

Lower bound of distribution

Upper bound of distribution



Selecting Probability Query Types
Using the Continuous Uniform Calculator
Calculating Exponential Distribution Probabilities
Working with Normal Distribution Calculator
Understanding Range Queries P(x₁ < X < x₂)
Reading PDF Curve Visualizations
What Are Continuous Probability Distributions?
PDF vs CDF for Continuous Distributions
Choosing the Right Continuous Distribution
Related Probability Tools and Calculators






Selecting Probability Query Types

Each continuous distribution calculator offers six query modes for comprehensive probability analysis. Full distribution displays the complete probability density function (PDF) curve showing the distribution's shape across its domain. The curve's height represents density, not probability—probability comes from area under the curve.

P(X < x) and P(X ≤ x) calculate left-tail probabilities, the area under the PDF curve from negative infinity (or minimum) up to x. For continuous distributions, P(X < x) equals P(X ≤ x) since probability at exact points is zero. The visualization shades the left region, making cumulative probability visible.

P(X > x) and P(X ≥ x) compute right-tail probabilities, the area from x to positive infinity (or maximum). These queries answer "What's the probability of exceeding x?" Both give identical results for continuous distributions. The shaded region shows probability graphically as area.

P(x₁ < X < x₂) calculates interval probability, the area between two values. This query requires both lower bound x₁ and upper bound x₂. The calculator shades the region between these bounds, demonstrating that continuous probability equals area under the PDF curve. This mode suits questions like "What's the probability of landing between 2 and 5?"

Using the Continuous Uniform Calculator

The continuous uniform distribution models scenarios where all values in an interval are equally likely—random selection from a range. Enter a (minimum) as the lower bound and b (maximum) as the upper bound. For random time between 0 and 10 minutes, set a=0, b=10.

The PDF equals 1/(b-a), constant across the interval. The visualization displays a rectangle: flat top at height 1/(b-a), spanning from a to b, zero outside this range. This rectangular shape explains the "uniform" name—density is uniform (constant) throughout.

Mean equals (a+b)/2, the midpoint. For a=0, b=10, mean is 5. Variance follows ((b-a)²)/12, measuring spread. The larger the interval, the greater the variance. Standard deviation equals the square root of variance.

Query probabilities reduce to simple geometry. P(X < x) for x between a and b equals (x-a)/(b-a), the fraction of the interval below x. P(x₁ < X < x₂) equals (x₂-x₁)/(b-a), the fraction between bounds. The calculator shades these rectangular regions orange, making probability calculations visual.

Use continuous uniform for random arrival times in windows, random coordinates in bounded regions, or any scenario with constant probability density over a finite interval.

Calculating Exponential Distribution Probabilities

The exponential distribution models time between events in a Poisson process—waiting times for arrivals, failures, or occurrences at constant rate. Enter λ (rate parameter) as the average event rate. For λ=0.5 events per minute, expect one event every 2 minutes on average.

The PDF equals λe^(-λx) for x≥0, an exponential decay curve starting at height λ when x=0, decreasing asymptotically toward zero. The visualization shows this characteristic decay: probability density highest near zero (short waits most likely), declining for longer waits.

Mean equals 1/λ, the average waiting time. For λ=0.5, mean is 2 minutes. Variance equals 1/λ², and standard deviation equals 1/λ, same as the mean. This property—mean equals standard deviation—uniquely characterizes exponential distributions.

The memoryless property distinguishes exponential distributions: P(X > s+t | X > s) = P(X > t). Past waiting doesn't affect future probability. If you've waited 5 minutes, probability of waiting 2 more equals the original 2-minute wait probability.

Query results use CDF = 1 - e^(-λx). For P(X < 3) with λ=0.5, calculate 1 - e^(-1.5) ≈ 0.7769. The shaded area shows cumulative probability. Use exponential for customer service times, equipment lifetimes, radioactive decay intervals, or any memoryless waiting time scenario.

Working with Normal Distribution Calculator

The normal (Gaussian) distribution is the most important continuous distribution, modeling countless natural and social phenomena. Enter μ (mean) as the distribution center and σ (standard deviation) as the spread measure. For μ=100, σ=15 (like IQ scores), the distribution centers at 100 with typical spread of 15 points.

The PDF produces the famous bell curve: symmetric around μ, highest at the mean, declining smoothly on both sides. The visualization shows this bell shape with mean marked by a red dashed line. Standard deviation marks appear at μ±σ, showing where density drops to about 60% of peak.

The 68-95-99.7 rule provides quick probability estimates: approximately 68% of values fall within μ±σ, 95% within μ±2σ, 99.7% within μ±3σ. For μ=100, σ=15, about 68% of values lie between 85 and 115.

When querying specific values, the calculator computes z-scores: z = (x-μ)/σ, measuring how many standard deviations x is from the mean. For x=115, μ=100, σ=15, z-score is 1. The result displays both probability and z-score. Standard normal tables use z-scores to find probabilities.

Use the range query for interval probabilities. P(-1 < Z < 1) for standard normal (μ=0, σ=1) gives approximately 0.68, confirming the 68% rule. The shaded region between bounds visualizes this central probability.

Apply normal distribution to heights, test scores, measurement errors, or any variable resulting from many independent factors. The central limit theorem ensures many real phenomena approximate normality.

Understanding Range Queries P(x₁ < X < x₂)

Range queries calculate probability of landing within a specified interval, fundamental for confidence intervals and tolerance limits. Select the range option, then enter both x₁ (lower bound) and x₂ (upper bound). The calculator computes P(x₁ < X < x₂) as the area between these bounds.

For continuous uniform distributions, range probability equals (x₂-x₁)/(b-a), simple fraction of the total interval. If the distribution spans 0 to 10 and you query 2 to 5, probability is 3/10 = 0.3. The rectangular shaded region shows this proportion visually.

For exponential distributions, use CDF values: P(x₁ < X < x₂) = CDF(x₂) - CDF(x₁) = [1-e^(-λx₂)] - [1-e^(-λx₁)]. With λ=0.5, P(1 < X < 3) equals CDF(3) - CDF(1) ≈ 0.7769 - 0.3935 = 0.3834. The shaded area under the exponential curve shows this probability.

For normal distributions, range queries use the error function (erf) through CDF calculations: P(x₁ < X < x₂) = Φ((x₂-μ)/σ) - Φ((x₁-μ)/σ) where Φ is the standard normal CDF. For μ=0, σ=1, P(-1 < X < 1) ≈ 0.6827, the 68% rule interval. The bell curve shading between -1 and 1 visualizes this central probability.

Range queries answer practical questions: "What fraction of products measure between spec limits?", "What percentage of customers wait 2-5 minutes?", "How likely is a value within tolerance?" The visual shading reinforces that continuous probability equals area under the density curve.

Reading PDF Curve Visualizations

The probability density function (PDF) curve visualization demonstrates each distribution's characteristic shape and how probability relates to area. The horizontal axis shows the random variable x, vertical axis shows density f(x). Remember: height is not probability—area under the curve between two points gives probability.

For continuous uniform, the PDF displays as a flat rectangle starting at x=a, ending at x=b, with constant height 1/(b-a). Zero density appears outside [a,b]. Total area under this rectangle equals 1, confirming it's a valid PDF. Query shading highlights rectangular regions whose areas equal the desired probabilities.

For exponential, the PDF curve starts at maximum height λ when x=0, then decays exponentially, approaching zero as x increases. The curve never touches zero but gets arbitrarily close. The red dashed line at x=μ=1/λ marks the mean, dividing the distribution into regions containing roughly 63% (left) and 37% (right) of probability.

For normal, the symmetric bell curve peaks at x=μ with maximum density 1/(σ√(2π)). Standard deviation markers at μ±σ indicate inflection points where curvature changes. The curve extends infinitely in both directions, though density becomes negligible beyond ±4σ. Symmetry means median equals mean.

Shaded regions in orange indicate the area representing your query's probability. For P(X<x), shading extends from the left edge to x. For P(X>x), shading extends from x rightward. For range queries, shading fills the interval between x₁ and x₂. These visual aids reinforce that probability = area under the PDF curve.

What Are Continuous Probability Distributions?

Continuous probability distributions describe random variables that can take any value in an interval or range, not just discrete points. Unlike discrete distributions assigning probability to specific values, continuous distributions use probability density across intervals. Examples include time, distance, weight, temperature—any measured quantity.

The probability density function (PDF) f(x) specifies density at each point, but f(x) itself is not probability. Instead, probability comes from integrating the PDF: P(a < X < b) = ∫[a to b] f(x)dx, the area under f(x) from a to b. The PDF must satisfy two properties: f(x) ≥ 0 everywhere, and total area ∫[-∞ to ∞] f(x)dx = 1.

A crucial property: probability at exact points equals zero for continuous distributions. P(X=x) = 0 for any specific x because a point has no width, hence no area. Consequently, P(X<x) = P(X≤x) and P(X>x) = P(X≥x)—inequalities with or without equality give identical results. Only intervals have nonzero probability.

Key statistics include mean μ = ∫xf(x)dx (expected value), variance σ² = ∫(x-μ)²f(x)dx (spread measure), and standard deviation σ (square root of variance). These integrals weight values by their densities, producing distribution centers and spreads.

The cumulative distribution function (CDF) F(x) = P(X≤x) accumulates probability from -∞ to x. CDF is the integral of PDF: F(x) = ∫[-∞ to x] f(t)dt. CDF increases from 0 to 1, with derivative F'(x) = f(x) recovering the PDF.

For comprehensive theory on continuous probability distributions including derivations, properties, and applications, see continuous probability distributions theory.

PDF vs CDF for Continuous Distributions

The probability density function (PDF) f(x) and cumulative distribution function (CDF) F(x) are complementary representations of continuous distributions. PDF shows density—how probability is distributed—while CDF shows accumulated probability up to each point.

PDF f(x) gives density at x, but NOT probability at x (which is always zero for continuous distributions). To get probability, integrate: P(a < X < b) = ∫[a to b] f(x)dx. The calculator visualizations show PDF curves with shaded areas representing probabilities. PDF can exceed 1 (it's density, not probability), but total area under PDF must equal 1.

CDF F(x) = P(X ≤ x) accumulates probability from -∞ to x. For each point x, CDF gives the probability of being at or below that point. CDF ranges from 0 to 1, always non-decreasing. At x=-∞, F(x)=0; at x=∞, F(x)=1. The calculators compute CDF values to answer queries like P(X<x) or P(X>x).

Relationship: CDF is the integral of PDF: F(x) = ∫[-∞ to x] f(t)dt. Conversely, PDF is the derivative of CDF: f(x) = F'(x). This fundamental relationship connects density and cumulative probability. Interval probabilities use CDF: P(a < X < b) = F(b) - F(a).

For uniform distribution, PDF is constant 1/(b-a) on [a,b], CDF is linear increasing from 0 to 1. For exponential, PDF is λe^(-λx), CDF is 1-e^(-λx). For normal, PDF is the bell curve, CDF is the S-shaped curve symmetric around μ with inflection at the mean.

The CDF visualizer tools display cumulative probability curves complementing these PDF visualizations, showing how probability accumulates across the distribution's range.

Choosing the Right Continuous Distribution

Selecting the appropriate continuous distribution requires matching scenario characteristics to distribution properties. Continuous Uniform applies when all values in a bounded interval are equally likely with no preference. Use for random selections from ranges, arrival times in windows, coordinates in bounded regions. The flat PDF indicates no value is more likely than others.

Exponential distribution models waiting times or intervals between events occurring at constant rate. Critical property: memoryless—past doesn't affect future probability. Use for customer service times, equipment failure times, time between arrivals, radioactive decay. If rate is constant and process has no memory, exponential fits. Single parameter λ controls both mean and spread (both equal 1/λ).

Normal (Gaussian) distribution models variables resulting from many independent additive factors. The central limit theorem ensures sums of random variables approach normality. Use for heights, weights, test scores, measurement errors, natural phenomena. Symmetric bell curve, mean=median=mode, fully characterized by μ and σ. When variables cluster around a central value with symmetric spread, normal fits.

Other continuous distributions (not in this calculator): Log-normal for positive variables that are multiplicative (stock prices, incomes), Gamma for sums of exponential variables (total time for k events), Beta for proportions bounded between 0 and 1, Weibull for failure times with changing hazard rates, Chi-square and t-distributions for statistical inference.

Selection criteria: Consider range (bounded vs unbounded), symmetry (normal is symmetric, exponential skewed), tail behavior (exponential has long right tail, normal has thin tails), memoryless property (only exponential), central limit effects (many independent factors suggest normal). Plot data and compare to distribution shapes.

Related Probability Tools and Calculators

Discrete Distributions Calculator - Compute probabilities for binomial, Poisson, geometric, and other discrete distributions with PMF charts.

Z-Score Calculator - Convert between raw scores and z-scores for normal distributions with percentile lookups.

Probability Distribution Explorer - Interactive visualization comparing multiple distributions side-by-side with parameter adjustments.

CDF Visualizer - Generate cumulative distribution function plots for discrete and continuous distributions.

Statistical Distribution Reference - Comprehensive guide to all probability distributions with formulas, properties, and use cases.

Confidence Interval Calculator - Compute confidence intervals using normal, t, and other distributions.

Central Limit Theorem Simulator - Visualize how sample means approach normal distribution regardless of population distribution.

Percentile Calculator - Find percentiles and quantiles for normal and other continuous distributions.

Normal Probability Plot Generator - Create Q-Q plots to assess whether data follows a normal distribution.

Hypothesis Testing Tools - Perform statistical tests using normal, t, chi-square, and F distributions.