Question

Suppose that the random variables X1, . . . , Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1, . . . , Xn}, and let Yn = max{X1, . . . , Xn}. Find E(Y1) and E(Yn).

Answer 1

To find E(Y1), integrate the PDF of Y1 from 0 to 1. The PDF of Y1 is n(1-x)^(n-1). To find E(Yn), integrate the PDF of Yn from 0 to 1. The PDF of Yn is n(x^n-1).

Step-by-step explanation:

The random variables X1, ..., Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. To find E(Y1), we need to find the expected value of Y1, which is the minimum of the X variables.

Since the X variables are uniformly distributed on [0, 1], the probability density function (PDF) for each X is 1, and the cumulative distribution function (CDF) is x for 0 <= x < 1. So, to find E(Y1), we integrate the PDF of Y1 from 0 to 1. The PDF of Y1 is n(1-x)^(n-1). Integrating this from 0 to 1 gives us E(Y1) = 1/(n+1).

To find E(Yn), we need to find the expected value of Yn, which is the maximum of the X variables. The CDF of the maximum of n X variables is the nth power of the CDF of a single X variable.

So, for the uniform distribution on [0, 1], the CDF is x for 0 <= x < 1. Therefore, the CDF of Yn is x^n for 0 <= x < 1. To find E(Yn), we integrate the PDF of Yn from 0 to 1. The PDF of Yn is n(x^n-1). Integrating this from 0 to 1 gives us E(Yn) = n/(n+1).