Pick a calculator with the three buttons above the card: Continuous Uniform, Exponential or Normal; each keeps its own values. Learn more about switching calculators
Pick a Probability Type: the full distribution, P(X < x), P(X ≤ x), P(X > x), P(X ≥ x) or the range P(x₁ < X < x₂); changing it clears the result and the x boxes. Learn more about the query types
For a query other than the full distribution, type x, or x₁ and x₂ for the range, in the boxes that appear below the parameters. Learn more about range queries
Click Calculate; it stays grey until every box is valid. The query card shows the probability to six decimals (with the z-score for the normal), then the cards for the mean, the variance and the standard deviation. Learn more about the results
Read the PDF curve under the cards: the shaded area is the probability you asked for, the dashed line the mean. Learn more about the curve
Read the panel on the right: it explains the active calculator and the chosen query type, with links to the sections below. Learn more about the normal calculator
Click Reset to empty the boxes of the active tool and remove its result; the other tools are left as they were. Learn more about resetting
The row of buttons above the calculator holds the three tools of this page: Continuous Uniform, Exponential and Normal Distribution Calculator. The page opens on the uniform calculator; click a button to switch, and the blue heading of the card names the active tool. Each calculator keeps its own boxes, so the values you typed into one are not carried into another, and the result of the tool you leave is dropped when you come back to it.
Adding ?tab=2 or ?tab=3 after the page address opens the exponential or the normal calculator directly, in the order of the buttons; the buttons themselves do not change the address. Every tool has the same layout: the Probability Type box, its parameter boxes, the Calculate and Reset buttons, the result cards and the PDF curve with the shaded area of the query. The panel on the right of each tool explains the active calculator and the chosen query type. Which tool fits which situation is explained in Choosing the Right Continuous Distribution.
Selecting Probability Query Types
DemoThree query types on the uniform distribution
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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?" Changing the type clears the result and empties the x boxes, so type the value again and click Calculate. The six types are compared in PDF vs CDF for Continuous Distributions, and each has its own section, starting with The Full Distribution Query.
Using the Continuous Uniform Calculator
DemoThe interval 0 to 10: full distribution, P(X < 3), b below a
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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. Both boxes take decimals and b must exceed a ("b must be greater than a"). For a=0, b=10 the cards read PDF 0.1000, mean 5.0000, variance 8.3333, standard deviation 2.8868; P(X<3)=0.3000 and P(2<X<5)=0.3000. The formulas are explained in The Continuous Uniform Distribution.
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. The λ box takes a positive decimal ("Lambda must be positive") and the x boxes non-negative values. For λ=0.5 the cards read mean 2.0000, variance 4.0000, standard deviation 2.0000; P(X<3)≈0.7769, P(X>2)≈0.3679 and P(1<X<3)≈0.3834. The formula is explained in The Exponential Distribution.
Working with Normal Distribution Calculator
Demoμ = 100, σ = 15: the bell curve, P(X < 115), σ refused
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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. The σ box must be positive ("Standard deviation must be positive"). For μ=100, σ=15 the cards read mean 100.0000, variance 225.0000, standard deviation 15.0000; P(X<115)≈0.8413 with the z-score 1.0000 shown under the probability, and P(85<X<115)≈0.6827. The formula is explained in The Normal Distribution.
Understanding Range Queries P(x₁ < X < x₂)
DemoTwo ranges on the normal distribution, then x₂ below x₁
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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. The range type adds a second box; x2 must exceed x1 ("x2 must be greater than x1"). The query is explained in Between Two Values: P(x₁ < X < x₂) with the three distributions as examples.
Reading PDF Curve Visualizations
DemoThe exponential curve and two shaded areas
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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. The curve belongs to the Full distribution type and the shaded area to the other five; what the area means is explained in What Are Continuous Probability Distributions?.
Calculating, Reading the Results and Resetting
DemoA refused pair, the result cards, Reset
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The Calculate button stays grey until every required box holds a valid value: the parameters, and the x box, or the two boxes x1 and x2 of the range type, for every query type except the full distribution. A value of the wrong kind is flagged under its box as you type or when you click, for example "Lambda must be positive", "Standard deviation must be positive", "b must be greater than a" or "x2 must be greater than x1", and the result stays empty until the note is cleared.
A successful click fills the right-hand side. For a query type other than the full distribution, the first card shows the query in symbols, such as P(X<3), with its probability to six decimals; the normal calculator adds the z-score under it. Then the parameter cards follow: the mean μ, the variance σ2 and the standard deviation σ in every tool, plus the constant density f(x) for the uniform tool. Distribution Info repeats the PDF formula, and the curve below shades the area of the query. Reset empties every box of the active tool and removes its result; the query type and the other two tools are left as they were. What the mean and variance of a continuous distribution measure is explained in What Are Continuous Probability Distributions?.
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.
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. Each of the three distributions has its own section on this page: The Continuous Uniform Distribution, The Exponential Distribution and The Normal Distribution; the lesson hub compares them in What they share and How they differ.
The Continuous Uniform Distribution
The continuous uniform distribution spreads probability evenly over an interval [a,b]: the density is the constant f(x)=b−a1 inside the interval and 0 outside, so the PDF is a rectangle of height b−a1 and area 1. Probabilities are lengths: P(X≤x)=b−ax−a for x in the interval and P(x1<X<x2)=b−ax2−x1.
The mean is the midpoint 2a+b and the variance 12(b−a)2. For a=0, b=10: density 0.1, mean 5, variance 12100≈8.3333, standard deviation ≈2.8868, P(X<3)=0.3 and P(2<X<5)=0.3. The common mistake is to read the density 0.1 as a probability; a single value has probability 0, and only an interval has a positive probability.
The exponential distribution models the waiting time until the next event when events occur at a constant average rate λ. Its density is f(x)=λe−λx for x≥0, a curve that starts at height λ and decays, and its CDF has the closed form P(X≤x)=1−e−λx, so every query of the calculator is one exponential away: P(X>x)=e−λx and P(x1<X<x2)=e−λx1−e−λx2.
The mean is λ1 and so is the standard deviation; the variance is λ21. For λ=0.5: mean 2, variance 4, P(X<3)=1−e−1.5≈0.7769, P(X>2)=e−1≈0.3679 and P(1<X<3)≈0.3834. The distribution is memoryless: having waited s already does not change the distribution of the remaining wait. The common mistake is to enter the mean waiting time as λ; the rate is its reciprocal, λ=1/mean.
The normal distribution is the bell curve with centre μ and spread σ: f(x)=σ2π1e−(x−μ)2/(2σ2). Its CDF has no closed form, so the calculator evaluates it numerically; the standard trick is the z-score z=σx−μ, the distance from the mean in standard deviations, which the calculator prints under every query probability.
For μ=100, σ=15: the variance is 225, P(X<115)=Φ(1)≈0.8413 since 115 is one standard deviation above the mean, and P(85<X<115)≈0.6827, the first part of the 68-95-99.7 rule. The common mistake is to enter the variance in the σ box; the box takes the standard deviation, so 225 would spread the curve fifteen times too wide.
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.
Full distribution is the opening query type of every calculator and needs only the parameters. Calculate then draws the PDF curve over a sensible range, marks the mean with a dashed line, and shows the parameter cards: the uniform rectangle of height b−a1, the exponential decay starting at λ, or the bell curve peaking at μ. No x box is shown in this mode.
Use it first to see the shape of the distribution and where the mean sits: for the exponential the mean λ1 lies to the right of the peak, for the normal it is the peak, for the uniform it is the midpoint. The area under the whole curve is 1, which is the second axiom of a PDF. The PDF and its rules are taught in The two fundamental axioms and the three ways to show it in Representations: the three views.
Strictly Less Than: P(X < x)
P(X < x) is the area under the density to the left of x, the cumulative distribution function F(x). For a continuous variable a single point carries no probability, so P(X<x) and P(X≤x) are the same number; the calculator offers both labels for readers coming from the discrete case, where they differ. For the uniform distribution on [0,10], P(X<3)=0.3; for the exponential with λ=0.5, P(X<3)=1−e−1.5≈0.7769; for the normal with μ=100, σ=15, P(X<115)≈0.8413.
The shaded region runs from the left edge of the curve to x. The common mistake is to expect a different answer from P(X≤x), as for a discrete variable; the difference, P(X=x), is 0 here. The function is defined in The definition of the CDF and its continuous case in The CDF of a continuous variable.
At Most: P(X ≤ x)
P(X ≤ x) is the cumulative distribution function F(x) itself, the area to the left of and up to x. It equals P(X<x) for every continuous distribution, since the boundary point adds no area; both types shade the same region and return the same six decimals. For the exponential distribution the formula is explicit, F(x)=1−e−λx; for the uniform it is linear, b−ax−a; for the normal it is Φ(σx−μ), evaluated numerically.
This is the query behind "at most", "no more than" and "within the first" questions, and the one whose values a textbook table lists. Every other query type can be written with it: 1−F(x) for the right tail and F(x2)−F(x1) for an interval. See The definition of the CDF and Using the CDF.
Strictly Greater Than: P(X > x)
P(X > x) is the right tail, the area to the right of x, equal to 1−F(x). For the exponential distribution it has the simplest form of all, P(X>x)=e−λx: with λ=0.5, P(X>2)=e−1≈0.3679, the chance that the wait lasts beyond 2. For the uniform distribution on [0,10], P(X>3)=0.7; for the normal with μ=100, σ=15, P(X>115)≈0.1587.
The shaded region runs from x to the right edge of the curve. The common mistake is to compute the tail by integrating the density from x to some finite bound; the calculator uses the complement 1−F(x), which is exact. Tails and complements are covered in Calculations using the PDF and Connection to CDF.
At Least: P(X ≥ x)
P(X ≥ x) is the right tail including x, 1−F(x) as well, since the point x adds no area for a continuous variable; it returns the same number as P(X>x) and shades the same region. The two labels are kept so that "at least" and "more than" questions can be entered as they are worded.
The query is the one behind "at least", "no less than" and "survives beyond" questions, for example the survival probability of a component whose lifetime is exponential: with λ=0.5 per year, P(X≥2)≈0.3679. The common mistake is to expect it to exceed P(X>x) by P(X=x), as in the discrete case; here P(X=x)=0. The relation to the CDF is taught in Connection to CDF and The CDF of a continuous variable.
Between Two Values: P(x₁ < X < x₂)
P(x₁ < X < x₂) is the area between two values, F(x2)−F(x1). The range type shows two boxes and needs x2>x1. For the uniform distribution on [0,10] the answer is a length ratio, P(2<X<5)=103=0.3; for the exponential with λ=0.5, P(1<X<3)=e−0.5−e−1.5≈0.3834; for the normal with μ=100, σ=15, P(85<X<115)=Φ(1)−Φ(−1)≈0.6827.
This is the query behind tolerance limits and confidence intervals: the share of parts within specification, the fraction of customers served within a window. The common mistake is to add the two one-sided probabilities instead of subtracting the CDF values. Interval probabilities are taught in Calculations using the PDF and Using the CDF.
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.