Why does a uniform distribution always have a rectangular graph?

Conditions

• The distribution is uniform.
• Applies to both discrete and continuous uniform distributions.

Reasoning, step by step

1. Recall the definition of a uniform distribution, where each event is equally likely.
2. Understand that equal likelihood translates to a constant probability density or mass.
3. Visualize a graph with a constant height over an interval, which forms a rectangular shape.

Example

The video shows a generic rectangular graph with a flat top at height $\frac{1}{b-a}$ spanning from $a$ to $b$, and a bar chart for rolling a die with six bars of equal height $\frac{1}{6}$, illustrating that the constant probability creates a rectangular or flat-topped shape.

Common misconceptions

• Assuming the graph shape depends on the specific values of the bounds rather than the constant probability.
• Confusing the rectangular shape of a uniform distribution with the bell shape of a normal distribution.

Watch the explanation

Explore next

• What is the probability density function formula for a continuous uniform distribution on the interval [a, b]?
• How do you draw the graph of a continuous uniform distribution?
• What is a real-world example of a continuous uniform distribution?