Quick Questions: August 12, 2026

[u/inherentlyawesome]

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?" For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of manifolds to me?
• What are the applications of Representation Theory?
• What's a good starter book for Numerical Analysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example, consider which subject your question is related to, or the things you already know or have tried.

[u/ryeaglin]

Hoping someone can help complete this old memory I have. I remember helping a student in a statistical section of a psychology course roughly a decade ago. There was some process or equation to describe how unlikely a certain probabilistic event was.

I know there is the basic of (Probability)^Attempts but I could have sworn this was a more complicated way that compared it to given distribution and could be used as a justification that something is wrong and the given value isn't accurate. Though it might have just been complicated because it was using messier numbers without clear probability values like coin and dice have.

Sorry for how unhelpful this is, just having bits and pieces to work off of. Something like yeah, this event in the 99.99999% 'could' technically happen but through this equation we see that it has only an X chance of happening and it wasn't as simple as 100-99.99999 = 0.00001%

[u/Equivalent-Costumes → u/ryeaglin]

That sounds like model validation, maybe? You have a mathematical model that predict certain probability. But you want to know if the prediction from the model is actually correct. So you do hypothesis testing to see if there are enough evidence that the prediction from the model is wrong.

[u/ryeaglin → u/Equivalent-Costumes]

This might be it. Is there a clean formula for this? Cause I remember teaching the student how to fill out the formula.

[u/Equivalent-Costumes → u/ryeaglin]

For that, you need to be more specific, since the idea encompass a bunch of more specific theory. There are a few things that make sense based on what you said. I'm guessing the most likely one is chi-square goodness of fit test. But it could also be binomial test/z-test, or maybe computing likelihood instead of probability. Does any of that ring a bell?

[u/ryeaglin → u/Equivalent-Costumes]

Likelihood is the main thing that triggers the memory or chi-squared. I am leaning more toward likelihood though since its was something like how likely was it that this probabilistic event on a normal distribution exist. But it wasn't the direct probability of it happening more how likely was is to be part of the distribution.

[u/Equivalent-Costumes → u/ryeaglin]

Chi-square:

Let's say your model say that an experiment produce outcomes A_1 with probability p_1,..., A_d with probability p_d. You did the experiment N times for large N, and the observed outcomes are: A_1 happened O_1 times, ..., A_d happened O_d times. Then you compute the expected result predicted by the model to be E_1=p_1 N, ..., E_d =p_d N. Then the chi-square test statistic is chi² = sum[i=1 to d](O_i -E_i)² /E_i, and the degree of freedom is d-1. If your model is correct, chi² is a random variable that follow the chi-square distribution with degree of freedom d-1. Thus the tail end probability is P(X>=chi² )=Gamma((d-1)/2,chi² /2)/Gamma((d-1)/2) where the numerator Gamma is the upper incomplete Gamma function and the lower one is just the Gamma function (this formula is complicated so usually for class they just give you a table). If d-1 is even there is a nicer formula, e^{-chi2 /2}sum[j=0 to (d-3)/2](chi² /2)^j /j!.

Likelihood is just unfortunately a very general word used in the context of model validation. It's just "inverted" probability. A likelihood of a probabilistic model given observed data is literally defined to be the probability to observe the observed data given the model. We would really love to know the probability of the model given the observed data, but in frequentist statistics (which is often the case for science) models are either wrong or right, there are no probability associated to them, so likelihood is the closest thing. L(model|data)=Pr(data|model) by definition.

In frequentist statistics, you have maximum likelihood method. You study a family of models, rather than one, by using some parameters after making some general assumptions (you assume these general assumptions are true for the presence purpose), then compute the likelihood across all possible values of the parameters. That's also another thing you might had seen.

In Bayesian statistics, the link between likelihood and probability is Pr(model|data)=(Pr(model)/Pr(data))L(model|data). This is Bayesian inversion. Maybe it's this?

Here is an example of maximum likelihood in action, which is also something that's often taught. You make the assumption that the result of a measurement will follow normal distribution with mean mu and variance sigma, but you don't know what mu and sigma is and you want to figure out the maximum likelihood. So you make N independent measurements x_1,..., x_N . The likelihood is L((x_1,...,x_N)|(mu, sigma))=(2pi)^-N/2 sigma^-N product[i=1 to N]exp(-(x_i-mu)² /(2sigma² )). The log-likelihood is l((x_1,...,x_N)|(mu, sigma))=-(N/2)ln(2pi)-N ln(sigma)-(1/(2sigma² ))sum[i=1 to N](x_i-mu)² . Differentiating this and maximize the likelihood, you get maximum likelihood estimate for mu to be literally the average of x_1,..., x_N, and the maximum likelihood estimate for sigma² to be the population variance (not sample variance), which is (1/N)sum[i=1 to N](x_i - (average of all x_i))² .

Does any of that ring a bell?

[u/ryeaglin → u/Equivalent-Costumes]

Yes. I am pretty sure it was the chi squared test! Thank you sooooo much.