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Let $X_1,X_2,\ldots,X_n$ be a random sample from a continuous distribution with the probability density function $$f(x)=\frac{1}{2\sqrt{2\pi}}\left[e^{-\frac12(x-2\mu)^2}+e^{-\frac12(x-4\mu)^2}\right],\quad-\infty<x<\infty$$

If $T=\sum\limits_{i=1}^n X_i$, then which one of the following is an unbiased estimator of $\mu$?

$$(A)\quad\frac{T}{n}\qquad\quad(B)\quad\frac{T}{2n}\qquad\quad(C)\quad\frac{T}{3n}\qquad\quad(D)\quad\frac{T}{4n}$$

If a function has a PDF that is the linear combination of two normal distribution, how to find its unbiased estimator?

I know the unbiased estimator of normal distribution is $\mu$ but in linear combination I don't know.

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  • $\begingroup$ Please use MathJax for typesetting math here. $\endgroup$ – StubbornAtom Oct 2 '19 at 14:53
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The key is to recognize that if $Z_i$ is a Bernoulli($1/2$) random variable, and the conditional distribution of $X_i$ given $Z_i=z$ is $N(2\mu, 1)$ if $z=0$, and $N(4\mu, 1)$ if $z=1$, then the marginal PDF of $X_i$ is the given density $f$.

Using the tower property, we have $\mathbb{E}[X_i] = \mathbb{E}[\mathbb{E}[X_i \mid Z_i]] = \frac{1}{2} (2 \mu) + \frac{1}{2} (4\mu) = 3 \mu$. From here you can compute $\mathbb{E}[T]$.

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