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Okay, so I have a n-dimensional normal distribution with the means and co-variance matrix defined (for now we can assume that these are the true distribution parameters, not estimates). For a given data point I want to calculate the probability that this point belongs to this distribution.

I believe I would be interested in the probability of generating a point "at least as unlikely" as the given data point. In a 1D normal distribution case this would be the area under the "two tails" of the PDF. E.g. 1-(CDF(x)-CDF(mu-x)). Now I want to calculate this probability for a general n-dimensional normal distribution. I believe maybe this should be 1 - "volume inside the hyper-sphere in Mahalanobis distance space defined by the radius given by the Mahalanobis distance of the data point from the mean".

Question 1: Is this interpretation correct?

Question 2: How do I calculate this?

Question 3: Does the analysis change if the mean and covariance matrix are only estimates of the true parameters?

Question 4: Is there an easy way to do this is python?

My best attempt at solving this question myself (but my statistics knowledge isn't very good, so I am asking for correctness confirmation, it is very important I get this correct) According to wikipedia:(https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Cumulative_distribution_function) So I can first calculate the Mahalanobis distance as above (MD), and then maybe I just have to calculate the CDF of the chi-squared distribution at MD, and take 1 minus this.

I have no idea if this is correct, but currently my best guess.

Thanks in advance.

Okay, so I have a $n$-dimensional normal distribution with the means and co-variance matrix defined (for now we can assume that these are the true distribution parameters, not estimates). For a given data point I want to calculate the probability that this point belongs to this distribution.

I believe I would be interested in the probability of generating a point "at least as unlikely" as the given data point. In a $1D$ normal distribution case this would be the area under the "two tails" of the PDF. E.g. $1-(CDF(x)-CDF(\mu-x))$. Now I want to calculate this probability for a general $n$-dimensional normal distribution. I believe maybe this should be $1$ - "volume inside the hyper-sphere in Mahalanobis distance space defined by the radius given by the Mahalanobis distance of the data point from the mean".

Question 1: Is this interpretation correct?

Question 2: How do I calculate this?

Question 3: Does the analysis change if the mean and covariance matrix are only estimates of the true parameters?

Question 4: Is there an easy way to do this is python?

My best attempt at solving this question myself (but my statistics knowledge isn't very good, so I am asking for correctness confirmation, it is very important I get this correct) According to wikipedia:

$$(x-\mu)^T\Sigma^{-1}(x-\mu) \le \chi_k^2(p)$$

So I can first calculate the Mahalanobis distance as above (MD), and then maybe I just have to calculate the CDF of the chi-squared distribution at MD, and take $1$ minus this.

I have no idea if this is correct, but currently my best guess.

Thanks in advance.

Okay, so I have a n-dimensional normal distribution with the means and co-variance matrix defined (for now we can assume that these are the true distribution parameters, not estimates). For a given data point I want to calculate the probability that this point belongs to this distribution.

I believe I would be interested in the probability of generating a point "at least as unlikely" as the given data point. In a 1D normal distribution case this would be the area under the "two tails" of the PDF. E.g. 1-(CDF(x)-CDF(mu-x)). Now I want to calculate this probability for a general n-dimensional normal distribution. I believe maybe this should be 1 - "volume inside the hyper-sphere in Mahalanobis distance space defined by the radius given by the Mahalanobis distance of the data point from the mean".

Question 1: Is this interpretation correct?

Question 2: How do I calculate this?

Question 3: Does the analysis change if the mean and covariance matrix are only estimates of the true parameters?

Question 4: Is there an easy way to do this is python?

My best attempt at solving this question myself (but my statistics knowledge isn't very good, so I am asking for correctness confirmation, it is very important I get this correct) According to wikipedia:(https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Cumulative_distribution_function) So I can first calculate the Mahalanobis distance as above (MD), and then maybe I just have to calculate the CDF of the chi-squared distribution at MD, and take 1 minus this.

I have no idea if this is correct, but currently my best guess.

Thanks in advance.

Okay, so I have a n-dimensional normal distribution with the means and co-variance matrix defined (for now we can assume that these are the true distribution parameters, not estimates). For a given data point I want to calculate the probability that this point belongs to this distribution.

I believe I would be interested in the probability of generating a point "at least as unlikely" as the given data point. In a 1D normal distribution case this would be the area under the "two tails" of the PDF. E.g. 1-(CDF(x)-CDF(mu-x)). Now I want to calculate this probability for a general n-dimensional normal distribution. I believe maybe this should be 1 - "volume inside the hyper-sphere in Mahalanobis distance space defined by the radius given by the Mahalanobis distance of the data point from the mean".

Question 1: Is this interpretation correct?

Question 2: How do I calculate this?

Question 3: Does the analysis change if the mean and covariance matrix are only estimates of the true parameters?

Question 4: Is there an easy way to do this is python?

My best attempt at solving this question myself (but my statistics knowledge isn't very good, so I am asking for correctness confirmation, it is very important I get this correct) According to wikipedia:(https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Cumulative_distribution_function) So I can first calculate the Mahalanobis distance as above (MD), and then maybe I just have to calculate the CDF of the chi-squared distribution at MD, and take 1 minus this.

I have no idea if this is correct, but currently my best guess.

Thanks in advance.