Final answer:
To find the probability that the range of a sample of size 4 from the uniform distribution is less than 1/2, it is required to integrate the joint probability density function over the appropriate range, considering the properties of the uniform distribution.
Step-by-step explanation:
The question asks to calculate the probability that the range of a random sample of size 4 from the uniform distribution is less than 1/2. The uniform distribution is denoted as U(a, b), where all values between a and b are equally likely, and the probability density function (PDF) for the uniform distribution on the interval (0,1) is f(x) = 1 for 0 < x < 1.
To find the probability that the range, defined as Yn - Y1 for a sample of size n, is less than 1/2, we can utilize the properties of the uniform distribution where Y1 is the smallest value and Yn is the largest value in the ordered sample.
Since Y1 and Yn are independent, we can define a new variable R = Yn - Y1 which represents the range. We want to find P(R < 1/2). This involves integrating the joint probability of Y1 and Yn over the region where their difference is less than 1/2, while considering the possible values they can take given the uniform distribution constraints.
By setting up and evaluating the integral, we calculate the desired probability. However, it requires certain calculations and understanding of probability integrals and the structure of uniform distributions, which is beyond the scope of a simple answer. This would typically involve integrals and considerations of the density function for the uniform distribution.