Question

Let X1,⋯,Xn be an iid random sample from a Uniform (0,θ), where θ>0. Let X(n)=max{X1,⋯,Xn} (a) Find the mean and variance of X(n) . (b) Find the limiting distribution of Zn=n(θ−X(n)). (c) Find the limiting distribution of log(Zn).

Answer 1

To find the mean and variance of X(n), we first need to determine the distribution of X(n) by finding its cumulative distribution function (CDF) and differentiating it to get the probability density function (PDF). The mean of X(n) is nθ/(n+1) and the variance of X(n) is nθ^2 / [(n+2)(n+1)] - (nθ / (n+1))^2. The limiting distribution of log(Zn) is a normal distribution, according to the Central Limit Theorem.

Step-by-step explanation:

To find the mean and variance of X(n), we first need to determine the distribution of X(n). Since X1,⋯,Xn are iid random samples from a Uniform (0,θ) distribution, we can determine the distribution of X(n) by finding its cumulative distribution function (CDF) and then differentiating it to get the probability density function (PDF).

1. The probability distribution of X(n) is given by:
F(x) = (x/θ)^n

2. Differentiating the CDF, we get the PDF:
f(x) = (n/θ)(x/θ)^(n-1)

(a) To find the mean of X(n), we integrate the PDF:
E[X(n)] = ∫x * f(x) dx = nθ / (n+1)

(b) To find the variance of X(n), we use the formula:
Var[X(n)] = E[X(n)^2] - (E[X(n)])^2

First, we calculate the second moment:
E[X(n)^2] = ∫x^2 * f(x) dx = nθ^2 * (n+1) / ((n+2)(n+1)^2)

Then, we substitute the values into the variance formula to get:
Var[X(n)] = nθ^2 / [(n+2)(n+1)] - (nθ / (n+1))^2

(c) To find the limiting distribution of log(Zn), we can use the Central Limit Theorem. As n approaches infinity, the distribution of Zn converges to a standard normal distribution. Taking the logarithm of Zn converts it to a log-normal distribution, which means that the limiting distribution of log(Zn) is a normal distribution.