Normal density function is:
If the random variable X obey the mathematical expectation of a [mu], and variance [sigma] 2 normal distribution, denoted by N ([mu], [sigma] 2 ). When = 0 [mu], [sigma] 2 =. 1 is referred to as a standard normal distribution. This does not need to remember complex formulas, you can know its meaning, when in use can access at any time.
When normal study, we believe that each sample is of equal weight, thus μ is the mean of random variable, control the position of the curve, [sigma] 2 controls the steepness of the curve:
[sigma] 2 is smaller, the closer the samples μ:
In the figure, when σ = 0.2, the steeper the curve, inverted bell narrower, more concentrated samples at the μ.
The maximum likelihood estimator
Normally distributed random variable X:
If there are n samples may be observed, according to the maximum likelihood function formula:
among them:
Logarithmic likelihood function, and based on the number of continuing simplification formula:
It can be learned by the ①:
We can now draw a final conclusion:
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