Suppose $x$ has a uniform distribution on the interval $[0,1]$. What is the probability that $x<0.25$?

• A. $0.5$
• B. $0.25$
• C. $0.75$
• D. $1$

Answer: (B)

Verified step by step guidance

1. Recognize that the random variable \(x\) follows a uniform distribution on the interval \([0, 1]\). This means that the probability density function (pdf) is constant over this interval.
2. Recall that for a uniform distribution on \([a, b]\), the pdf is given by \(f(x) = \frac{1}{b - a}\) for \(a \leq x \leq b\). Here, \(a = 0\) and \(b = 1\), so \(f(x) = 1\) for \$0 \leq x \leq 1$.
3. The probability that \(x\) is less than a certain value \(c\) within the interval \([0, 1]\) is the area under the pdf from \(a\) to \(c\). Mathematically, this is \(P(x < c) = \int_a^c f(x) \, dx\).
4. Substitute the values into the integral: \(P(x < 0.25) = \int_0^{0.25} 1 \, dx\).
5. Evaluate the integral to find the probability: \(P(x < 0.25) = 0.25 - 0 = 0.25\). This is the probability that \(x\) is less than 0.25.