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Visible Probability: Continuous Distributions

Continuous | Discrete | Custom

Here are some ideas for how to use this tool:

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Enter the parameters of the distribution you wish to analyze. Drag the left and right limits of integration (the dark gray box) to compute the probability that a random variable following that distribution will be within those limits. Change to one of the "family" modes to get a better understanding of how the shape of the distribution depends on the parameters. Use the sampling tools at the bottom of the page to understand how random varibles' values depend on the shape of the distribution, and to learn about confidence intervals.
You can explore discrete distributions by clicking on 'Discrete' above.

Pick a probability distribution: μ = σ = λ = μ = σ = k = θ = x₀ = γ = k = ν = d₁ = d₂ = k = λ = θ = α = β = [Save] [Load]

Interpretation: X is the position of a particle after a large number (say n) steps of 1-dimensional Brownian motion, starting at position μ, with steps of size σ / √n. X is the amount of time between events in a memoryless process with rate λ. X is the amount of time before k events occur in memoryless process with rate 1 / θ. X is the ratio of a normal random variable with mean μ and standard deviation σ to an independent normal random variable with mean 0 and standard deviation 1. X is sum of the squares of k independent normal random variables with mean 0 and standard deviation 1. X is the mean of a set of ν+1 random samples from a normal population with mean 0 and standard deviation 1. X is d₂ / d₁ times the ratio of two χ² variables with parameters d₁ and d₂. X is the expected lifetime of something, where k controls death rate as a function of time. X is the α'th highest of α+β-1 uniformly picked random numbers between 0 and 1.

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Min:	Integrate from:	to:	Max:
p(?) = ?

This curve shows how likely a random variable with the given distribution is to take on any particular value. Places where the curve is taller correspond to more likely values. The area under the curve between the two dotted lines represents the total probability that a random sample taken from this distribution will lie between these limits.

You can change the parameters of the distribution (e.g. μ and σ for the Normal distribution) and see how they affect the shape of the PDF. If you click on the popup menu that says "Integrate" and change it to one of the family modes, you can see curves for several different parameter values at once. The blue curves have larger parameter values, while the red ones have smaller ones.

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How many samples?
Sample μ: ? Sample σ: ?

Try clicking the "Sample" button to pick 10 random values from this distribution. Each value will show up as a short red line. The mean of the samples will be plotted as a tall blue line. The sample mean should be roughly equal to the mean of the distribution (aka the "population mean"), but it will not usually be exactly equal. Try clicking the "sample" button several more times to get a feel for how far off the sample mean will typically be from the mean of the distribution.

The distribution that describes how likely the sample mean is to be a particular distance from the true mean is called the sampling distribution of the mean. If you are sampling from a normal distribution, it is a nontrivial fact that that sampling distribution of the mean with N samples has the shape of the Student t-distribution with k = N-1. Using this fact, it is possible to compute a so-called 95% confidence interval for the mean based on a sample. Shown as a blue rectangle in sample above, it has the property that for 95% of samples taken, the corresponding confidence interval will contain the true mean.