Stage 1: Understanding Random Variables and Distributions

1. Select a distribution type (Uniform, Normal, or Exponential) from the dropdown menu.

2. Adjust the distribution parameters using the sliders:

   • For Uniform: Set the minimum and maximum values
   • For Normal: Set the mean and standard deviation
   • For Exponential: Set the rate parameter

3. Set the number of points (100-1000) using the slider.

4. Click "Generate Data" to create a new dataset.

5. Observe the generated points on the plot and the basic statistics (mean and standard deviation) in the information panel.

6. Repeat steps 1-5 with different parameter values to understand how they affect the distribution.

Stage 2: Analyzing Sample Size Effects

1. Choose a distribution type and set its parameters.

2. Generate datasets with different sample sizes (e.g., 100, 300, 500, 700, 1000).

3. For each sample size:

   • Generate multiple datasets
   • Record the mean and standard deviation
   • Note how these statistics vary with sample size

4. Compare the observed statistics with the theoretical values:

   • For Uniform: mean = (min + max)/2, variance = (max - min)²/12
   • For Normal: mean = μ, variance = σ²
   • For Exponential: mean = 1/λ, variance = 1/λ²

5. Draw conclusions about how sample size affects the accuracy of statistical estimates.

Stage 3: Interactive Data Generation and Analysis

1. Use the "Animate" button to see how the points are generated in real-time.

2. Try different combinations of:

   • Distribution types
   • Parameter values
   • Sample sizes

3. Observe how the shape of the distribution changes with different parameters.

4. Compare the theoretical distribution shape with the actual generated points.

5. Note how the basic statistics (mean and standard deviation) help characterize the distribution.

Learning Objectives:

• Understand different types of random variables and their distributions
• Learn how parameters affect the shape and characteristics of distributions
• Observe the relationship between sample size and statistical accuracy
• Develop intuition about probability distributions through interactive visualization
• Understand basic statistical measures (mean and standard deviation) in the context of different distributions