Basic Continuous Density Functions
Learning Objective
Use area under the curve to compute probability for continuous probability density functions.
Why This Matters
Every time Uber estimates "4 minutes to pickup," that prediction comes from a continuous probability model of arrival times -- not a lookup table of exact values. The same math powers ChatGPT's response-time monitoring, hospital triage systems, and the quality control that decides whether your phone's battery will last the advertised 10 hours. This simulation is where you learn why asking "what's the probability the wait is between 3 and 5 minutes" makes sense, but asking "what's the probability the wait is exactly 3.4782 minutes" gives you a surprising answer of zero.
How to Use This Simulation
- Choose a preset distribution to see different curve shapes -- start with the uniform distribution where area is just base × height.
- Drag the bound handles on the x-axis (or type values below the chart) and watch the shaded area change. The shaded area IS the probability.
- Switch shading modes (left tail, right tail, between) to compute different types of probability.
- Check the Explanation Panel below -- it updates as you interact and tells you exactly what's happening.
Step-by-Step Probability Calculation
Enter distribution parameters and bounds to see the probability calculation broken down step by step. For a uniform distribution, this is simple geometry. For a normal distribution, it requires the CDF.
Enter values above and click Calculate to see the step-by-step work.
What's Happening
Quick Check
A continuous random variable X follows a normal distribution. A student claims that P(2 < X < 5) must be slightly less than P(2 ≤ X ≤ 5) because the second expression "includes the endpoints." Is the student correct?
Try This
Select the "Bus Wait Times" preset (Uniform, 0 to 10 minutes). The density is constant at f(x) = 0.1 across the interval. Set the shading mode to "P(a < X < b)" and drag the lower bound to 3 and the upper bound to 7.
Before checking the simulation's answer, calculate P(3 < X < 7) by hand using base × height: (7 − 3) × 0.1. Does your answer match the simulation?
Switch to the "Standardized Scores" preset (Normal, μ = 0, σ = 1). Using the 68-95-99.7 rule, predict the approximate probability that a z-score falls between −1 and 1. Set the bounds to −1 and 1 and check your prediction.
Now calculate P(X > 2) using the right-tail shading mode. Explain in one sentence why you can't compute P(X > 2) for a normal distribution using simple base × height the way you did for the uniform distribution.
A customer service center's hold times follow a normal distribution with μ = 4 minutes and σ = 1.2 minutes. Use the simulation's μ and σ sliders (or the Step-by-Step Calculator tab) to find three probabilities:
(1) What percentage of callers wait less than 2 minutes? (2) What percentage waits longer than 6 minutes? (3) The company guarantees "most calls answered within 6 minutes." Based on your answer to question 2, write one sentence recommending whether the company should keep the guarantee, and name the probability that supports your recommendation.
Instructor Notes
Teaching Notes
This simulation bridges Sim 14's discrete probability to continuous probability. The most effective sequence: start with the uniform preset where P = base × height is geometrically obvious, then switch to the normal curve and let students discover that the same area-under-curve logic applies even though the calculation is no longer simple geometry. The aha moment comes when students drag both bounds to the same point and see the area collapse to exactly zero -- that's when P(X = a) = 0 clicks.
The "Sensor Precision" preset intentionally produces density values above 1. Let students encounter this and struggle with it before explaining that density is not probability. If you preempt the confusion, you rob them of the conceptual work.
Common Student Errors
- Reading the y-axis value as probability instead of density. Students who mastered discrete PMFs (where the y-axis IS probability) carry that intuition here.
- Believing that P(X = a) must be some very small positive number, not exactly zero. The limit argument is subtle -- the simulation's visual collapse is more convincing than the algebra.
- Thinking the total area can exceed 1 if the curve is wide, or that something is wrong if density exceeds 1. Both stem from conflating density with probability.
- Assuming "between" probability is symmetric around the mean. P(μ − 1 < X < μ) does not equal P(μ < X < μ + 2) even though both spans are 1-2 units wide.
Discussion Questions
- In Sim 14, each bar's height was the probability of that outcome. Why can't the curve's height be the probability here? What changed when we moved from discrete to continuous?
- If P(X = 3.0000) = 0 for a continuous distribution, how can we still observe the value 3.0000 in real data? (Hint: measurement precision.)
- A normal distribution with σ = 0.1 has a peak density near 4.0. Does this mean values near the mean are "400% likely"? What does density actually measure?
Exam Connection
Typical exam questions ask students to compute P(a < X < b) for a given normal distribution using z-tables. The Step-by-Step Calculator tab directly prepares students for this format by showing the z-transformation and CDF lookup. Some exams also test whether P(X < 3) equals P(X ≤ 3) for continuous distributions -- the Quick Check addresses this directly.