Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compute a convolution using overlap length.
- Justify the conclusion "fS(s)=s or 2−s on those intervals, and zero elsewhere" using the stated assumptions.
Before you start
Independent densities and integrals.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For independent X,Y uniform on (0,1), find the density of S=X+Y.
Why this math matters
Compute a convolution using overlap length. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The variables are independent.
- Both uniform intervals have unit length.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Derive a triangular distribution from two uniforms
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For independent X,Y uniform on (0,1), find the density of S=X+Y.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
fS(s)=∫1(0<x<1)1(0<s−x<1)dx
The convolution counts the overlap of two allowed intervals.
Work through the mathematics
The overlap length is s for 0<s<1 and 2−s for 1≤s<2
The support grows and then shrinks as the sum changes.
Check the conclusion
fS(s)=s or 2−s on those intervals, and zero elsewhere
The two triangular halves each have area one half, so the density normalizes.
The result
fS(s)=s or 2−s on those intervals, and zero elsewhere
The two triangular halves each have area one half, so the density normalizes.
Common mistakes to catch
- The sum is not uniform on (0,2).
- The integration limits depend on the requested sum.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find P(S≤1).
Show a hint
Integrate s from zero to one.
Reveal answer and explanation
1/2
The area under the first half of the triangle is one half.
Practice 2
What is Var(S)?
Show a hint
Add independent uniform variances.
Reveal answer and explanation
1/6
Each uniform has variance 1/12.
Take the idea with you
Explain why sums of bounded independent timing errors cluster near the middle.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Put an average on the central-limit scale
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