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2. The random variable X has the following probability density function: \begin{equation} f(x) = \frac{2}{\lambda^2}xe^{-x^2/\lambda^2}, \quad x \ge 0 \end{equation} where $\lambda$ is a positive parameter. (15 points) (a) It can be shown that \begin{equation} E(X) = \frac{\sqrt{\pi\lambda}}{2} \end{equation}and \begin{equation} E(X^2) = \lambda^2 \end{equation} Set up (but do not evaluate) the definite integrals that are needed to show this. (b) Let $X_1, X_2, ..., X_n$ be a random sample from X. Find an unbiased estimator $\hat{\lambda}$ for $\lambda$ of the form (some constant) $\cdot \overline{X}$ (c) Find V($\hat{\lambda}$). (d) Is $\hat{\lambda}$ a consistent estimator for $\lambda$? Explain.

          2. The random variable X has the following probability density function:
\begin{equation*}
f(x) = \frac{2}{\lambda^2}xe^{-x^2/\lambda^2}, \quad x \ge 0
\end{equation*}
where $\lambda$ is a positive parameter.
(15 points)
(a) It can be shown that
\begin{equation*}
E(X) = \frac{\sqrt{\pi\lambda}}{2}
\end{equation*}and
\begin{equation*}
E(X^2) = \lambda^2
\end{equation*}
Set up (but do not evaluate) the definite integrals that are needed to show this.
(b) Let $X_1, X_2, ..., X_n$ be a random sample from X. Find an unbiased estimator
$\hat{\lambda}$ for $\lambda$ of the form
(some constant) $\cdot \overline{X}$
(c) Find V($\hat{\lambda}$).
(d) Is $\hat{\lambda}$ a consistent estimator for $\lambda$? Explain.

2. The random variable X has the following probability density function:

f(x) = (2)/(λ^2)xe^-x^2/λ^2, x ≥ 0

where λ is a positive parameter.
(15 points)
(a) It can be shown that

E(X) = (√(πλ))/(2)
and

E(X^2) = λ^2

Set up (but do not evaluate) the definite integrals that are needed to show this.
(b) Let X1, X2, ..., Xn be a random sample from X. Find an unbiased estimator
λ̂ for λ of the form
(some constant) ·X
(c) Find V(λ̂).
(d) Is λ̂ a consistent estimator for λ? Explain.

Added by Eric A.

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