How do you calculate probability with continuous distributions?

Calculate probability by integrating the PDF over the desired interval. Use P(a ≤ X ≤ b) = ∫[a to b] f(x)dx. Alternatively, use the CDF: P(a ≤ X ≤ b) = F(b) - F(a), where F is the cumulative distribution function.

Why is the probability of an exact value zero?

In continuous distributions, there are infinitely many possible values. The probability of any single exact value is zero because a point has zero width. Probabilities only make sense over intervals with positive width, calculated through integration.

What's the difference between PDF and CDF?

PDF (probability density function) shows relative likelihood and requires integration to find probabilities. CDF (cumulative distribution function) directly gives P(X ≤ x) without integration. The CDF is the integral of the PDF from negative infinity to x.

When should you use a Normal distribution?

Use Normal distribution for measurements affected by many independent random factors, such as heights, weights, test scores, and errors. The Central Limit Theorem makes Normal appropriate for sample means. Normal fits symmetric data without heavy tails or extreme outliers.

Continuous Probability Distribution Visualizer

Interactive visualization of fundamental continuous distributions

Continuous Uniform

Constant probability over an interval

Lower Bound (a): 0

Upper Bound (b): 10

Explanation

The continuous uniform distribution has constant probability density over the interval $[a, b]$ . The probability density function is $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$ , and $0$ otherwise. The expected value is $E[X] = \frac{a+b}{2}$ and the variance is $\text{Var}(X) = \frac{(b-a)^2}{12}$ . This distribution models situations where all values in an interval are equally likely, such as the position of a randomly thrown dart on a board, random arrival times within a time window, or measurement errors uniformly distributed within tolerances. Learn more about the continuous uniform density

Selecting a Distribution Type

Using the PDF vs CDF Toggle

Adjusting Distribution Parameters

Continuous Uniform: a Flat Density and a Straight CDF

Normal: the Bell and the Sigmoid

Exponential: Peak at Zero, Tail Without End

Reading the Curve

Understanding Density vs Probability

Comparing Distribution Shapes

What is a Probability Density Function?

Normal Distribution Fundamentals

Exponential Distribution Basics

Understanding Integration and Area

When to Use Each Distribution

Related Probability Concepts

Selecting a Distribution Type

Click the distribution tabs at the top to switch between three continuous distributions: Uniform, Normal (Gaussian), and Exponential. The selected distribution appears highlighted in blue. Each distribution has a brief description explaining its key characteristics. The Uniform distribution shows constant density over an interval. The Normal distribution displays the classic bell curve shape, symmetric around its mean. The Exponential distribution models waiting times with exponential decay. The tool maintains your previous parameter settings when switching between distributions, making it easy to compare how different distributions behave with similar parameter ranges.

Using the PDF vs CDF Toggle

Toggle between PDF (Probability Density Function) and CDF (Cumulative Distribution Function) views using the buttons above the chart. The PDF view shows the density curve, indicating relative likelihood at each point. Higher curves mean greater density, though the height itself is not a probability. The CDF view shows cumulative probability, displaying

P(X \leq x)

at each point. The CDF always increases from 0 to 1, creating an S-shaped curve for most distributions. Switch between views to understand how density accumulates into probability. The PDF shows where probability is concentrated, while the CDF shows total probability up to any point. Both views use the same parameters, so changes in one immediately reflect in the other.

Adjusting Distribution Parameters

Use the sliders to adjust parameters specific to each distribution. For the Uniform distribution, set the lower bound

a

and upper bound

b

to define the interval. The Normal distribution uses mean

\mu

to shift the curve left or right, and standard deviation

\sigma

to control spread. Larger

\sigma

creates wider, flatter curves; smaller

\sigma

creates taller, narrower curves. The Exponential distribution uses rate parameter

\lambda

, where larger values create steeper decay and smaller expected values. Drag sliders smoothly or click to jump to specific values. Parameter values display next to each label. The curve updates instantly as you adjust parameters, providing immediate visual feedback on how parameters affect distribution shape.

Continuous Uniform: a Flat Density and a Straight CDF

With

a = 0

and

b = 10

the PDF view is a flat plateau at

f(x) = \frac{1}{b-a} = 0.1

between the bounds and zero outside. Switch the toggle and the CDF view is a straight line of slope

0.1

rising from

0

1

.

The two stills below are the same distribution under the two toggle positions.

Continuous uniform on [0, 10]: PDF above, CDF below

A flat density of 0.1 and a straight-line CDF of slope 0.1. The CDF slope IS the PDF height, and its two corners are where the density jumps from 0 to 0.1 and back.

Seeing them together makes the relationship concrete: the CDF's slope *is* the PDF's height. A flat density gives a straight-line CDF, and the two corners in the CDF are exactly where the density jumps from

0

0.1

and back.

Notice that the density value

0.1

is not a probability.

P(X = 3) = 0

here, as it is for any continuous variable. What

0.1

means is probability per unit of

x

, so the probability of landing in an interval of width

2

0.2

— you have to multiply by a width before you get a probability at all.

Normal: the Bell and the Sigmoid

\mu = 0

and

\sigma = 1

the PDF view peaks at

f(0) = \frac{1}{\sqrt{2\pi}} = 0.3989

, and the CDF view is the symmetric S-curve crossing

0.5

at the mean.

The tool plots the window

\mu \pm 4\sigma

Normal, mean 0, sigma 1: PDF above, CDF below

The bell peaks at 1/sqrt(2 pi) = 0.3989 and the sigmoid crosses 0.5 at the mean. Shrink sigma and the peak rises without limit - a density may exceed 1, a probability may not.

The peak height is worth reading carefully, because it is the clearest illustration that a density can exceed any bound. Here it is

0.399

, comfortably under 1 — but shrink

\sigma

and the peak rises without limit, since the area must stay at 1. A density above 1 is perfectly legal; a probability above 1 is not.

The two views also locate

\sigma

differently. On the PDF it is the distance from the peak to the inflection points; on the CDF it is where the curve stops steepening. Same number, two quite different visual cues, which is the reason for having the toggle at all.

Exponential: Peak at Zero, Tail Without End

\lambda = 1

the PDF view starts at its maximum,

f(0) = \lambda = 1

, and decays; the CDF view rises steeply then flattens toward

1

.

The tool plots

x

from

0

out to

5/\lambda

Exponential, lambda = 1: PDF above, CDF below

Density is largest at the very edge of the support, f(0) = 1, with no interior peak. The plotted CDF ends at 0.9931 - the remaining 0.69% lies past the right edge of the window.

The exponential is the one distribution here whose density is largest at the very edge of its support. There is no interior peak: the most probable region is always immediately after zero, whatever

\lambda

is.

The right edge of the window is the detail to carry away. The plotted CDF ends at about

0.9931

, not

1

, and the missing

0.69\%

is spread over the infinite tail beyond. The PDF view makes the same point differently — the curve is visibly still above zero when the plot stops, so there is clearly more distribution to the right than the window shows.

Reading the Curve

The horizontal axis shows the random variable's values, while the vertical axis shows either density (PDF) or cumulative probability (CDF). For PDF views, curve height indicates relative likelihood. Higher sections correspond to more probable regions, but remember that height is density, not probability. Probability requires integrating the PDF over an interval. For CDF views, the curve height directly gives

P(X \leq x)

. At any point

x

, read up to the curve and across to the vertical axis to find the cumulative probability. Hover over the curve to see exact values in a tooltip. The tooltip displays both the

x

value and the corresponding function value (density or cumulative probability) to four decimal places.

Understanding Density vs Probability

Probability density is not the same as probability. Density can exceed 1, especially for narrow distributions concentrated in small intervals. The Normal distribution with small standard deviation can have peak densities much greater than 1. Probability is the area under the PDF curve over an interval, not the height. For any single point, probability equals zero in continuous distributions. To find probability for an interval

[a, b]

, integrate the PDF:

P(a \leq X \leq b) = \int_a^b f(x)dx

. The CDF provides this directly:

P(a \leq X \leq b) = F(b) - F(a)

. The total area under any PDF always equals 1, ensuring all possible outcomes have probability 1 collectively.

Comparing Distribution Shapes

Each distribution has distinctive shape characteristics. The Uniform distribution shows a flat, rectangular shape in PDF view, with constant density across the interval and zero outside. Its CDF is a straight diagonal line within the interval. The Normal distribution creates the famous bell curve in PDF view, symmetric and smooth with a single peak at the mean. Its CDF is an S-curve, transitioning smoothly from 0 to 1. The Exponential distribution shows exponential decay in PDF view, starting at its maximum and decreasing rapidly. Its CDF rises quickly then gradually approaches 1. Switch between distributions and adjust parameters to see how shapes transform. Understanding these shape patterns helps identify which distribution best models your data.

What is a Probability Density Function?

The probability density function (PDF) describes the relative likelihood of a continuous random variable taking values near each point. Written as

f(x)

, the PDF must be non-negative everywhere and integrate to 1 over its entire range. Unlike discrete PMFs that give exact probabilities, PDFs give density. Probability comes from integration:

P(a \leq X \leq b) = \int_a^b f(x)dx

. The height

f(x)

indicates relative likelihood but is not itself a probability. Regions with higher PDF have greater probability per unit width. For comprehensive theory on continuous random variables, integration, and cumulative distribution functions, see our detailed probability density function page.

Normal Distribution Fundamentals

The Normal distribution, also called Gaussian, is the most important continuous distribution. Its PDF is

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

. The parameters

\mu

(mean) and

\sigma

(standard deviation) determine location and spread. The distribution is perfectly symmetric around

\mu

. The 68-95-99.7 rule states that approximately 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. The Central Limit Theorem explains why Normal distributions appear throughout nature and statistics. For detailed coverage of z-scores, standardization, and applications in statistical inference, see our Normal distribution theory page.

Exponential Distribution Basics

The Exponential distribution models the time between events in a Poisson process. Its PDF is

f(x) = \lambda e^{-\lambda x}

for

x \geq 0

, where

\lambda

is the rate parameter. Both the expected value and standard deviation equal

1/\lambda

. The distribution has the memoryless property: the probability of an event in the next interval doesn't depend on how much time has passed. This makes it unique among continuous distributions. Common applications include modeling equipment lifetimes, waiting times, and radioactive decay. For comprehensive coverage of the Poisson process, memoryless property, and connections to reliability theory, see our Exponential distribution page.

Understanding Integration and Area

Integration calculates the area under the PDF curve, converting density into probability. The definite integral

\int_a^b f(x)dx

gives

P(a \leq X \leq b)

, the probability that

X

falls between

a

and

b

. For the entire range,

\int_{-\infty}^{\infty} f(x)dx = 1

, ensuring probabilities sum to certainty. The CDF is the integral of the PDF from negative infinity to

x

F(x) = \int_{-\infty}^{x} f(t)dt

. Most continuous distributions lack simple closed-form solutions for these integrals, requiring numerical methods. For detailed coverage of integration techniques, numerical integration, and CDF properties, see our comprehensive probability theory pages.

When to Use Each Distribution

Choose the Uniform distribution when all values in an interval are equally likely, such as random positions on a board or random timestamps within a window. Use the Normal distribution for measurements affected by many small independent factors, including heights, weights, test scores, and measurement errors. The Central Limit Theorem justifies Normal assumptions for sample means. Select the Exponential distribution for time until an event occurs, especially in memoryless scenarios like equipment failure or radioactive decay. If data shows decay that depends on history, consider other distributions like Weibull. For symmetric data with heavier tails than Normal, consider Student's t-distribution. Match your distribution choice to your data's shape and the process generating it.