If $X$ is a continuous random variable uniformly distributed on the interval $[0,4.5)$, what is the probability that $X<3$?

• A. $0.5$
• B. $\frac{4.5}{3}$
• C. $\frac{3}{4.5}$
• D. $1$

Answer: (C)

Verified step by step guidance

1. Identify the type of distribution: Since X is uniformly distributed on the interval [0, 4.5), it means the probability density function (pdf) is constant over this interval.
2. Recall the formula for the pdf of a continuous uniform distribution on [a, b): \(f(x) = \frac{1}{b - a}\) for \(a \leq x < b\).
3. To find the probability that \(X < 3\), calculate the length of the interval from the start of the distribution to 3, which is \$3 - 0 = 3$.
4. Use the formula for the probability over an interval in a uniform distribution: \(P(X < 3) = (3 - 0) \times f(x) = (3 - 0) \times \frac{1}{4.5 - 0}\).
5. Simplify the expression to get \(P(X < 3) = \frac{3}{4.5}\), which represents the probability that X is less than 3.