riaology
Unit 2 · Lesson 1 of 5

Probability is area beneath a density.

Waiting times can be modeled by a density curve. The chance of an event before a deadline is accumulated area, not curve height.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its interval probability.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase waiting time b? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Read the modelf(t)=λe−λt (t≥0),P(T≤b)=∫0bf(t) dt=1−e−λbf(t)=\lambda e^{-\lambda t}\ (t\ge0),\quad P(T\le b)=\int_0^bf(t)\,dt=1-e^{-\lambda b}

Exponential waiting-time model, nonnegative time, constant event rate λ.

Waiting time (min) versus Density (1/min). Density. Selected interval probability: 0.77687 . Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one.000.52.515Density (1/min)Waiting time (min)
Density

interval probabilityP(T≤3)=1−e−1.5=0.77687P(T\le3)=1-e^{-1.5}=0.77687

Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one.

Assumptions, units, and limits

Exponential waiting-time model, nonnegative time, constant event rate λ.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

01

From the situation to an equation

A density is nonnegative and integrates to one over its support. Integrate across an interval for probability. The cumulative distribution accumulates from the support's start; its derivative recovers the density where smooth.

f(t)=λe−λt (t≥0),P(T≤b)=∫0bf(t) dt=1−e−λbf(t)=\lambda e^{-\lambda t}\ (t\ge0),\quad P(T\le b)=\int_0^bf(t)\,dt=1-e^{-\lambda b}
02

Keep the reasoning visible

P(a<T<b)=∫abf(t) dtP(a<T<b)=\int_a^bf(t)\,dt

Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one.

03

What makes this model useful?

Exponential waiting-time model, nonnegative time, constant event rate λ.

Density can exceed one because it has inverse-time units. A single point has zero probability in this continuous model.

A worked example

Calculate the starting scenario

  1. Use waiting time b = 3 min; rate λ = 0.5 1/min.
  2. P(T≤3)=1−e−1.5=0.77687P(T\le3)=1-e^{-1.5}=0.77687
  3. Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one. The checked quantity is interval probability.
Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the interval probability using waiting time b 3 min, rate λ 0.5 1/min. State the meaning of its sign and units.

Checked quantity: interval probability · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

A density is nonnegative and integrates to one over its support. Integrate across an interval for probability. The cumulative distribution accumulates from the support's start; its derivative recovers the density where smooth.

Explore the visual solution
Read the modelf(t)=λe−λt (t≥0),P(T≤b)=∫0bf(t) dt=1−e−λbf(t)=\lambda e^{-\lambda t}\ (t\ge0),\quad P(T\le b)=\int_0^bf(t)\,dt=1-e^{-\lambda b}

Exponential waiting-time model, nonnegative time, constant event rate λ.

Waiting time (min) versus Density (1/min). Density. Selected interval probability: 0.77687 . Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one.000.52.515Density (1/min)Waiting time (min)
Density

interval probabilityP(T≤3)=1−e−1.5=0.77687P(T\le3)=1-e^{-1.5}=0.77687

Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one.

Assumptions, units, and limits

Exponential waiting-time model, nonnegative time, constant event rate λ.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

A density is nonnegative and integrates to one over its support. Integrate across an interval for probability. The cumulative distribution accumulates from the support's start; its derivative recovers the density where smooth.

f(t)=λe−λt (t≥0),P(T≤b)=∫0bf(t) dt=1−e−λbf(t)=\lambda e^{-\lambda t}\ (t\ge0),\quad P(T\le b)=\int_0^bf(t)\,dt=1-e^{-\lambda b}
Calculate and explain

Apply the idea to a new scenario

For rate λ=0.2 min⁻¹, find the probability that waiting time is at most 5 min.

Checked quantity: interval probability · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

A density is nonnegative and integrates to one over its support. Integrate across an interval for probability. The cumulative distribution accumulates from the support's start; its derivative recovers the density where smooth.

Explore the visual solution
Read the modelf(t)=λe−λt (t≥0),P(T≤b)=∫0bf(t) dt=1−e−λbf(t)=\lambda e^{-\lambda t}\ (t\ge0),\quad P(T\le b)=\int_0^bf(t)\,dt=1-e^{-\lambda b}

Exponential waiting-time model, nonnegative time, constant event rate λ.

Waiting time (min) versus Density (1/min). Density. Selected interval probability: 0.63212 . Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one.000.52.515Density (1/min)Waiting time (min)
Density

interval probabilityP(T≤5)=1−e−1=0.632121P(T\le5)=1-e^{-1}=0.632121

Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one.

Assumptions, units, and limits

Exponential waiting-time model, nonnegative time, constant event rate λ.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

A density is nonnegative and integrates to one over its support. Integrate across an interval for probability. The cumulative distribution accumulates from the support's start; its derivative recovers the density where smooth.

f(t)=λe−λt (t≥0),P(T≤b)=∫0bf(t) dt=1−e−λbf(t)=\lambda e^{-\lambda t}\ (t\ge0),\quad P(T\le b)=\int_0^bf(t)\,dt=1-e^{-\lambda b}
Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual
Read the modelf(t)=λe−λt (t≥0),P(T≤b)=∫0bf(t) dt=1−e−λbf(t)=\lambda e^{-\lambda t}\ (t\ge0),\quad P(T\le b)=\int_0^bf(t)\,dt=1-e^{-\lambda b}

Exponential waiting-time model, nonnegative time, constant event rate λ.

Waiting time (min) versus Density (1/min). Density. Selected interval probability: 0.77687 . Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one.000.52.515Density (1/min)Waiting time (min)
Density

interval probabilityP(T≤3)=1−e−1.5=0.77687P(T\le3)=1-e^{-1.5}=0.77687

Density height has units inverse time and is not probability at a point. Probability is area; total area across the support is one.

Assumptions, units, and limits

Exponential waiting-time model, nonnegative time, constant event rate λ.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. A density is nonnegative and integrates to one over its support. Integrate across an interval for probability. The cumulative distribution accumulates from the support's start; its derivative recovers the density where smooth.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

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