Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Differentiate a log-likelihood and check its maximum.
- Justify the conclusion "ℓ″(λ)=−3/λ²<0, so λ̂=1/2 is the unique maximum" using the stated assumptions.
Before you start
Exponential densities and derivatives.
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
Independent exponential observations are 1,2,3 time units. Find the maximum-likelihood rate estimate.
Why this math matters
Differentiate a log-likelihood and check its maximum. 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:
- Observations are positive, independent, and identically exponential.
- There is no censoring in the sample.
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.
Estimate a rate by maximizing its likelihood
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Independent exponential observations are 1,2,3 time units. Find the maximum-likelihood rate estimate.
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
L(λ)=λ³exp(−6λ), λ>0
Multiply the three exponential densities.
Work through the mathematics
ℓ′(λ)=3/λ−6=0 ⇒ λ̂=1/2
Taking logarithms simplifies the optimization without changing its maximizer.
Check the conclusion
ℓ″(λ)=−3/λ²<0, so λ̂=1/2 is the unique maximum
The estimate equals reciprocal sample mean, with rate measured per time unit.
The result
ℓ″(λ)=−3/λ²<0, so λ̂=1/2 is the unique maximum
The estimate equals reciprocal sample mean, with rate measured per time unit.
Common mistakes to catch
- A likelihood is a function of the parameter for fixed data.
- The MLE can be biased even when it is easy to compute.
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
What rate estimate comes from observations 2 and 6?
Show a hint
Use n divided by the total time.
Reveal answer and explanation
1/4
Two observations divided by total eight gives 0.25.
Practice 2
Is λ̂ the sample mean waiting time?
Show a hint
Check units and reciprocity.
Reveal answer and explanation
No
The sample mean estimates a time scale; λ̂ estimates its reciprocal rate.
Take the idea with you
Re-derive the likelihood if some waiting times are only known to exceed a threshold.
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: Estimate an endpoint by matching a theoretical moment
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.