Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Include a Jacobian when changing the variable of a density.
- Justify the conclusion "∫₀¹fY(y)dy=1 and E[Y]=1/3" using the stated assumptions.
Before you start
Inverse functions and differentiation.
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
If X is uniform on (0,1) and Y=X², find the density of Y.
Why this math matters
Include a Jacobian when changing the variable of a density. 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:
- X takes only positive values between zero and one.
- Density values outside (0,1) are zero.
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.
Transform a continuous random variable carefully
PausedQuestion: Start with the question. Paused.
Question
Start with the question
If X is uniform on (0,1) and Y=X², find the density of 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
FY(y)=P(X≤√y)=√y for 0<y<1
Monotonicity allows the event to be inverted.
Work through the mathematics
fY(y)=1/(2√y) on (0,1)
Differentiating the cumulative distribution supplies the change-of-variable factor.
Check the conclusion
∫₀¹fY(y)dy=1 and E[Y]=1/3
The density concentrates near zero but remains integrable, and ∫₀¹x²dx checks its mean.
The result
∫₀¹fY(y)dy=1 and E[Y]=1/3
The density concentrates near zero but remains integrable, and ∫₀¹x²dx checks its mean.
Common mistakes to catch
- Do not simply substitute √y into the old density.
- A density may diverge near an endpoint while still integrating to one.
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(Y≤1/4).
Show a hint
Use FY.
Reveal answer and explanation
1/2
X²≤1/4 corresponds to X≤1/2.
Practice 2
Would Y be uniform because X is uniform?
Show a hint
Compare equal-width y intervals under the inverse map.
Reveal answer and explanation
No
Squaring stretches intervals unevenly, changing their probability density.
Take the idea with you
Explain how squaring a uniformly chosen radius changes the distribution of squared radius.
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: Normalize probability on a triangular region
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