Question

(20pt) Suppose we have two independent random samples: X1.....Xn1 is an iid sample from EXP(θ₁) and Y1...Yn2 is an iid sample from EXP(θ₂) .consider Q=θ₂X/θ₁Y Prove Q is apivotal quantity and give its distribution.

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Answer 1

To prove that Q = θ₂X/θ₁Y is a pivotal quantity, we need to show that its distribution does not depend on the parameters θ₁ and θ₂. Q follows a non-central t-distribution with n = n₁ + n₂ - 2 degrees of freedom and a non-centrality parameter equal to (θ₁/θ₂) * (EX/EY), where EX and EY are the means of X and Y, respectively.

Step-by-step explanation:

To prove that Q = θ₂X/θ₁Y is a pivotal quantity, we need to show that its distribution does not depend on the parameters θ₁ and θ₂.

1. First, let's calculate the distribution of Q. We know that X ~ EXP(θ₁), so the probability density function (pdf) of X is given by f(x) = θ₁e^(-θ₁x).
2. Similarly, Y ~ EXP(θ₂), so the pdf of Y is f(y) = θ₂e^(-θ₂y).
3. Now, let's find the pdf of Q. We have Q = θ₂X/θ₁Y. Using the properties of exponents, we can simplify this to Q = θ₂/θ₁ * X/Y.
4. The ratio X/Y can be shown to follow a distribution called the ratio distribution, which has a non-central t-distribution with n = n₁ + n₂ - 2 degrees of freedom and a non-centrality parameter equal to (θ₁/θ₂) * (EX/EY), where EX and EY are the means of X and Y, respectively.
5. Since the ratio distribution does not depend on θ₁ and θ₂, we can conclude that Q is a pivotal quantity with a non-central t-distribution.