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I'm trying to add an extra bound of a gaussian probability distribution, solving for both $\mu$ and $\sigma$:

$L = f + \lambda(\int_{-\infty}^{\infty}p(x, \mu, \sigma)dx - 1)$ where $p$ is the expression for a normal distribution's probability.

I can't seem to wrap my head around how to approach this question, since I get stumped on the fact that it's a definite integral. Because it's definite, I can't just take the derivative of x and say it's done(or can I?). I also wanted to replace the pdf function with the cdf of the gaussian and use that as my constraint but it seemed too "unelementary", as I wanted something easier to handle.

Any ideas?

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    In your optimization is your restraint $\int_{-\infty}^{\infty}p(x)dx=1$? This is true by definition – 2017-01-24
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    Yes @Hugh , that's what I was suspecting, but then how come we need to take summations of bernoulli probabilities and make sure they sum to 1 as well? – 2017-01-24
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    It's not clear what you mean. The Bernoulli distribution also sums to one, that's a fundamental property of a probability distribution – 2017-01-24
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    Ah okay, so I'm trying to find the "maximum likelihood" of a dataset, and that pretty much means solving a maximization problem, where the likelihood is some probability. To make sure the axioms of probability hold, we can't just give every probability some bogus value to maximize the likelihood. That's why I needed the constraint on bernoulli before - in case the optimization problem caused the probabilities to be >1 or something similar to that. – 2017-01-24
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    You need to rethink what you're trying to achieve by restricting the sum of the pdf over all x. If you're optimizing for x then the sum of the pdf doesn't depend on x – 2017-01-24
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    Ah you're right, I'll edit the question. I meant to say the mean and the sigma of the gaussian equation. – 2017-01-24
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    As said in MLE you are maximzing over the parameters, which will be in terms of the observed random variables (data). And for any parametric family of distribution, like normal, the likelihood (joint pdf) will automatically satisfy all the necessary constriants for all parameters in the parameter space. So it is not necessary to add such constraint here. – 2017-01-25

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