A continuous random variable X is uniformly distributed with 0 ≤ X ≤ 20.
a. Draw a graph of the uniform density function.

Verified step by step guidance

1. Recall that for a continuous uniform distribution on the interval \([a, b]\), the probability density function (pdf) is constant and given by \(f(x) = \frac{1}{b - a}\) for \(a \leq x \leq b\), and \(f(x) = 0\) otherwise.
2. Identify the parameters for this problem: \(a = 0\) and \(b = 20\), so the pdf is \(f(x) = \frac{1}{20 - 0} = \frac{1}{20}\) for \$0 \leq x \leq 20$.
3. To draw the graph, plot the horizontal axis representing \(x\) from 0 to 20, and the vertical axis representing the density \(f(x)\).
4. Draw a horizontal line at the height \(f(x) = \frac{1}{20}\) between \(x = 0\) and \(x = 20\), indicating the constant density over this interval.
5. Outside the interval \([0, 20]\), the density is zero, so the graph should be on the \(x\)-axis (height 0) for \(x < 0\) and \(x > 20\).

Key Concepts

Uniform Distribution

A uniform distribution describes a continuous random variable where all outcomes in a specified interval are equally likely. For X uniformly distributed between 0 and 20, the probability density function (pdf) is constant over this range and zero elsewhere.

Probability Density Function (pdf)

The pdf of a continuous random variable shows the relative likelihood of the variable taking on a specific value. For a uniform distribution on [0, 20], the pdf is a horizontal line at height 1/(20-0) = 1/20, representing equal probability density across the interval.

Graphing the Density Function

To graph the uniform density function, plot the pdf as a constant horizontal line between 0 and 20 on the x-axis, with height equal to 1/20. Outside this interval, the pdf is zero, so the graph touches the x-axis. This visualizes the equal likelihood of all values in the range.