A continuous random variable X is uniformly distributed with 0 ≤ X ≤ 20.
b. What is P(0 ≤ X ≤ 5)?

Verified step by step guidance

1. Identify the type of distribution: Since X is uniformly distributed between 0 and 20, it means the probability density function (pdf) is constant over this interval.
2. Write down the pdf for a uniform distribution on the interval [a, b]: \(f(x) = \frac{1}{b - a}\) for \(a \leq x \leq b\). Here, \(a = 0\) and \(b = 20\), so \(f(x) = \frac{1}{20 - 0} = \frac{1}{20}\).
3. Understand that the probability for a continuous uniform distribution over an interval is the area under the pdf curve over that interval, which is the length of the interval times the height of the pdf.
4. Calculate the probability \(P(0 \leq X \leq 5)\) by multiplying the length of the interval (5 - 0) by the pdf value: \(P(0 \leq X \leq 5) = (5 - 0) \times \frac{1}{20}\).
5. Express the final probability as a simplified fraction or decimal after performing the multiplication.

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 a and b, the probability density function (pdf) is constant, and the total area under the pdf curve equals 1.

Probability for Continuous Random Variables

For continuous variables, probabilities are found by calculating the area under the probability density function over an interval. Since the pdf gives density, the probability of X falling within an interval [c, d] is the integral of the pdf from c to d.

Calculating Probability in Uniform Distribution

In a uniform distribution from a to b, the probability that X lies between c and d (where a ≤ c ≤ d ≤ b) is proportional to the length of the interval: P(c ≤ X ≤ d) = (d - c) / (b - a). This simplifies probability calculation to a ratio of interval lengths.