Final answer:
For a simple random sample from a Uniform(0,1) population, the expected value and variance of the sample median can be calculated using the formulas E(M) = (n+1)/(2n) and Var(M) = (n^2-1)/(12n^2(n+1)). For a sample size of 25, the expected value of the sample median is 0.52 and the variance is 0.04. The expected value of the sample midrange is 1, and the expected value of the sample range is 0.038.
Step-by-step explanation:
To find the expected value and variance of the sample median, we need to first understand the distribution of the sample median. In the case of a simple random sample from a Uniform(0,1) population, the sample median follows a Beta(0.5,n/2) distribution, where n is the sample size. The expected value of the sample median is given by E(M) = (n+1)/(2n), and the variance is given by Var(M) = (n^2-1)/(12n^2(n+1)). For a sample size of 25, the expected value of the sample median is E(M) = (25+1)/(2*25) = 13/25 = 0.52, and the variance is Var(M) = (25^2-1)/(12*25^2*(25+1)) = 24/600 = 0.04.
The sample midrange is the average of the smallest and largest observations in the sample. Since the observations are from a Uniform(0,1) population, the smallest and largest observations follow a Beta(1,n) distribution. The expected value of the sample midrange is given by E(R) = (1+n)/(n+1) = (1+25)/(25+1) = 26/26 = 1.
The sample range is the difference between the largest and smallest observations in the sample. Again, since the observations are from a Uniform(0,1) population, the smallest and largest observations follow a Beta(1,n) distribution. The expected value of the sample range is given by E(R) = 1/(n+1) = 1/(25+1) = 1/26 = 0.038.