Final answer:
The PDF of the minimum order statistic X1 from a sample with PDF fx(x;θ) = θ/x², x > 0, and θ > 0, can be calculated by obtaining the CDF of one observation, raising its complement to the power of n, and then differentiating it with respect to x.
Step-by-step explanation:
The student is asking for the probability density function (PDF) of the minimum order statistic X1 from a sample of n observations from a population with a given PDF fx(x;θ) = θ/x² for x > 0 and θ > 0. To find the PDF of the minimum order statistic, we use the fact that the cumulative distribution function (CDF) of X1 is the probability that all n observations are greater than a certain value x, which can be formalized as (1 - FX(x))n where FX(x) is the CDF of a single observation. Differentiating this with respect to x gives us the PDF of the minimum order statistic. Recall that the CDF is the integral of the PDF.
To provide a specific calculation, consider a population where the probability density function (PDF) of a random variable X is fx(x;θ) = θ/x². First, one would calculate the CDF of X, FX(x) = θ(1/x - 1/θ) for x ≥ θ. Then, raise the complement to the power of n to get the CDF for the minimum order statistic X1, (1 - FX(x))n. Finally, differentiate this CDF with respect to x to obtain the PDF of X1, which typically would result in the form fX1(x) = nθ(1/x²)(1 - θ(1/x - 1/θ))(n-1) for x within the specified domain.