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You differentiated with respect to $\sigma$, but $\sigma$ is fixed; your variables are $a$ and $b$, subject to $a+b=1$. So you need to differentiate $a^2\sigma^2+4b^2\sigma^2=a^2\sigma^2+4(1-a)^2\sigma^2=(5a^2-8a+4)\sigma^2$ with respect to $a$, yielding $10a-8=0$ and thus $a=\frac45$ as expected.


[by] "non-random parameter estimation" they are talking about the MLE. Is that right? They mean parameter as in parametric statistics, where the data are imagined to be random, but the type of random variation that led to that data is determined by some parameters such as the mean and variance of a Gaussian distribution. The parameters are thought of ...


First let's unpack the phrase "nonrandom parameter estimation". In context this means "estimating a parameter of a probability distribution for a random variable" where the parameter is a definite value which is only known through sampling of the random variable. For example, many repeated measurements may be performed in order to estimate the true value ...

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