Question

If $X_{1}, X_{2}, \ldots, X_{n}$ constitute a random sample of size $n$ from a geometric population, show that $Y=X_{1}+X_{2}+$ $\cdots+X_{n}$ is a sufficient estimator of the parameter $\theta$.

          If $X_{1}, X_{2}, \ldots, X_{n}$ constitute a random sample of size $n$ from a geometric population, show that $Y=X_{1}+X_{2}+$ $\cdots+X_{n}$ is a sufficient estimator of the parameter $\theta$.

Added by Lauren M.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Madhur L

04/21/2023

Step 1

The probability mass function (PMF) of a geometric distribution is given by: $$ P(X=k) = (1-\theta)^{k-1}\theta, \quad k=1,2,3,\ldots $$ where $0 < \theta < 1$ is the parameter of interest. Show more…

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If $X_{1}, X_{2}, \ldots, X_{n}$ constitute a random sample of size $n$ from a geometric population, show that $Y=X_{1}+X_{2}+$ $\cdots+X_{n}$ is a sufficient estimator of the parameter $\theta$.