Question

29. Suppose \{X_1, X_2, ..., X_{16}\} is a random sample coming from Gamma($\alpha = 4, \beta = 2$) distribution. Consider the following estimator for the population mean $\mu$: $\hat{\theta} = 3\bar{X} - 4X_2 + 2X_3$ (a) Show that $\hat{\theta}$ is an unbiased estimator for $\mu$. (b) Calculate the variance of $\hat{\theta}$, that is V($\hat{\theta}$). Show all work.

          29. Suppose \{X_1, X_2, ..., X_{16}\} is a random sample coming from Gamma($\alpha = 4, \beta = 2$) distribution.
Consider the following estimator for the population mean $\mu$:
$\hat{\theta} = 3\bar{X} - 4X_2 + 2X_3$
(a) Show that $\hat{\theta}$ is an unbiased estimator for $\mu$.
(b) Calculate the variance of $\hat{\theta}$, that is V($\hat{\theta}$).
Show all work.

29. Suppose {X1, X2, ..., X16} is a random sample coming from Gamma(α = 4, β = 2) distribution.
Consider the following estimator for the population mean μ:
θ̂ = 3X̅ - 4X2 + 2X3
(a) Show that θ̂ is an unbiased estimator for μ.
(b) Calculate the variance of θ̂, that is V(θ̂).
Show all work.

Added by Sarah R.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Kari Hasz

Step 1

- The mean of a Gamma distribution with parameters $\alpha$ and $\beta$ is $\mu = \alpha \beta$. - Here, $\alpha = 4$ and $\beta = 2$, so $\mu = 4 \times 2 = 8$. Show more…

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29. Suppose {X1, X2,..., X16} is a random sample coming from Gamma(a = 4, β = 2} distribution. Consider the following estimator for the population mean μ: $$\hat{\theta} = 3\overline{X} - 4X_2 + 2X_3$$ (a) Show that $\hat{\theta}$ is an unbiased estimator for μ. (b) Calculate the variance of $\hat{\theta}$, that is V($\hat{\theta}$). Show all work.