Probability Theory & Mathematical Statistics
Start gently
Collect enough of anything and the scatter settles into one fixed shape: the normal distribution. That property is why measuring a sample tells you about the whole.
Key points
Random variables and expected value
The expected value of a random variable X is its long-run average. For a continuous variable you compute it by integrating against the probability density function f(x).
The Central Limit Theorem
No matter what distribution your samples come from, the sample mean drifts toward a normal distribution as n grows. That is why so many real-world measurements end up looking bell-shaped — and it is what makes statistical inference work at all.
Maximum likelihood estimation (MLE)
A method for finding the parameter values that make the data you actually observed as unsurprising as possible. Training a machine learning model is usually written as maximizing the log-likelihood — which is the same thing as minimizing a loss function.
Bayesian updating
You start with a prior p(θ), fold in the evidence x, and end up with a posterior p(θ|x) — your updated belief about θ once x has been observed. Spam filters, medical diagnosis, and recommendation engines all run on this.
Monte Carlo methods
When an integral has no closed-form answer, you approximate it by drawing random samples instead. The same trick estimates π, prices financial derivatives, and powers Bayesian inference — hard math problems solved with the help of random numbers.
See it drawn
One die alone is flat. Add just one more and a peak appears. The more you add, the closer it gets to a normal distribution (the central limit theorem).
Jobs that use this
Statistician (academia and research institutes)$125k
Quantitative Analyst (financial engineering)$220k
Machine Learning Researcher$230k
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