Question

the values of a particular distribution are uniformly distributed between 2 and 7.

p(3 < x < 5) = \boxed{}

Explanation:

Step1: Recall Uniform Distribution Probability Formula

For a uniform distribution between \(a\) and \(b\), the probability density function is \(f(x)=\frac{1}{b - a}\) for \(a\leq x\leq b\), and the probability \(P(c < x < d)\) (where \(a\leq c < d\leq b\)) is given by \(P(c < x < d)=(d - c)\times\frac{1}{b - a}\). Here, \(a = 2\), \(b = 7\), \(c = 3\), \(d = 5\).

Step2: Calculate the Width of the Interval and the Total Interval

First, find the length of the interval \((3,5)\): \(d - c=5 - 3 = 2\). Then, find the length of the total interval \((2,7)\): \(b - a=7 - 2 = 5\).

Step3: Compute the Probability

Using the formula, \(P(3 < x < 5)=(5 - 3)\times\frac{1}{7 - 2}=\frac{2}{5}=0.4\).

Answer:

\(0.4\) (or \(\frac{2}{5}\))