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Undergraduate · Uniform convolution

Uniform convolution: Two narrow uniforms

Uniform convolution: investigate two narrow uniforms with first uniform width a = 0.5; second uniform width b = 0.5.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

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Uniform convolution: Two narrow uniforms. Sum value: 0. Density at that value: 0. Cumulative probability: 0. Expected sum: 0.5Probability comes from integrated density0200.51probability densitysum x → · labeled axes rescale to this model
The two variables are independent continuous uniforms with positive widths. The formula uses positive parts. The graph is an exact density with total area one, not a histogram or a claim that an exact point has positive probability.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Sum value
0
Density at that value
0
Cumulative probability
0
Expected sum
0.5

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Understand what you are seeing

The idea behind the motion.

The density of the sum of two independent uniforms is proportional to the overlap of two intervals. Equal widths give a triangle; unequal widths create a flat middle section. The density height is not a point probability, and integrating it gives the cumulative probability. This investigation starts with First uniform width a = 0.5; Second uniform width b = 0.5. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

f(x)=[x₊−(x−a)₊−(x−b)₊+(x−a−b)₊]/(ab)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Specify independent X on [0,a] and Y on [0,b]. The sum must lie between zero and a+b. The starting case is “Two narrow uniforms.”

  2. STEP 2

    Follow the changing quantity

    Move through possible sum values. Compare rising overlap, any plateau, and falling overlap with the density graph.

  3. STEP 3

    Explain and test the result

    Read cumulative probability at the marker and check that it reaches one at a+b. Compare the mean with (a+b)/2.

Your turn to explain

Make a prediction. Test your reasoning.

Keep First uniform width a = 0.5; Second uniform width b = 0.5. Pause the timeline at 100%. Given sum value = 1, calculate density at that value, cumulative probability, expected sum. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

At x=1, overlap length is max(0,min(x,a,b,a+b−x))=0. Divide by ab=(0.5)(0.5)=0.25 to get density 0. Integrating this piecewise-linear density through x gives cumulative probability 1. Results: Density at that value: 0; Cumulative probability: 1; Expected sum: 0.5. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Undergraduate · Statistics

Derive a triangular distribution from two uniforms

Compute a convolution using overlap length.