Consider a random sample X1, . . . , Xn from the uniform distribution over the interval (μ −sqrt(3)σ, μ + sqrt(3)σ).Let X(1) ≤ · · · ≤ X(n) denote the corresponding order statistics.(a) Find the mean and variance of X(n) − X(1)HINT: It might be easier to solve the problem for U1, . . . , Un, a random sample from the uniform distribution over either (0, 1) or (−1, 1), and then make an appropriate transformation.
Question
Consider a random sample X1, . . . , Xn from the uniform distribution over the interval (μ −sqrt(3)σ, μ + sqrt(3)σ).Let X(1) ≤ · · · ≤ X(n) denote the corresponding order statistics.(a) Find the mean and variance of X(n) − X(1)HINT: It might be easier to solve the problem for U1, . . . , Un, a random sample from the uniform distribution over either (0, 1) or (−1, 1), and then make an appropriate transformation.
Solution
The question is asking for the mean and variance of the range of a sample from a uniform distribution. The range is defined as the difference between the maximum and minimum values in the sample.
Here's how to solve it:
Step 1: Define the random variables Let's define the random variables U1, ..., Un as a random sample from the uniform distribution over (0, 1). Then, we can define the random variables X1, ..., Xn as X = μ + sqrt(3)σU - sqrt(3)σ/2, which is a transformation of U.
Step 2: Find the mean and variance of U The mean of a uniform distribution over (0, 1) is 0.5, and the variance is 1/12.
Step 3: Find the mean and variance of X Using the transformation X = μ + sqrt(3)σU - sqrt(3)σ/2, we can find that the mean of X is μ and the variance is σ^2.
Step 4: Find the mean and variance of the range The range of a sample from a uniform distribution is the difference between the maximum and minimum values. The expected value of the range for a uniform distribution over (0, 1) is (n-1)/2n, and the variance is (n-1)(n+1)/12n^3.
Step 5: Transform the results back to the original scale Using the transformation X = μ + sqrt(3)σU - sqrt(3)σ/2, we can find that the expected value of the range of X is (n-1)sqrt(3)σ/2n and the variance is (n-1)(n+1)σ^2/12n^3.
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