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The common distributions

Bernoulli, binomial, Poisson, uniform, exponential, normal, and what each models.

Binomial counts successes; Poisson counts rare events; exponential waits for the next one; normal is what sums look like. Picture it: the bell curve and the 95% between ±1.96. Think it: the normal appears because of the central limit theorem, not because nature prefers it.

Àwọn ààtò àwọn ìṣàmúlò-ètò · integrate e^(-x^2/2)/sqrt(2*pi) dx from -1.96 to 1.96

Calculus · Definite-Integrals

Integrate sqrt(2)·e^(-x^2/2)/(2·sqrt(pi)) from -1.96000 to 1.96000

\int_{-1.96}^{1.96} \frac{\sqrt{2} e^{- \frac{x^{2}}{2}}}{2 \sqrt{\pi}}\, dx

Àwọn ìṣàmúlò-ètò

\int_{-1.96}^{1.96} \frac{\sqrt{2} e^{- \frac{x^{2}}{2}}}{2 \sqrt{\pi}}\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
\int \frac{\sqrt{2} e^{- \frac{x^{2}}{2}}}{2 \sqrt{\pi}}\, dx = \frac{\sqrt{2}}{2 \sqrt{\pi}} \int e^{- \frac{x^{2}}{2}}\, dx
Pull the constant \frac{\sqrt{2}}{2 \sqrt{\pi}} out of the integral.
\int e^{- \frac{x^{2}}{2}}\, dx = \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}
Erf rule.
F(1.96) - F(-1.96) = \left(\frac{\operatorname{erf}{\left(0.98 \sqrt{2} \right)}}{2}\right) - \left(- \frac{\operatorname{erf}{\left(0.98 \sqrt{2} \right)}}{2}\right)
Fundamental theorem of calculus: plug in the limits.
= \operatorname{erf}{\left(0.98 \sqrt{2} \right)} \approx 0.95000
Simplify.

Fi àwọn àgbèwọlé hàn

\operatorname{erf}{\left(0.98 \sqrt{2} \right)}

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Bii o ṣe le: The common distributions

First find an antiderivative F, then evaluate F(b) − F(a).
Pull the constant \frac{\sqrt{2}}{2 \sqrt{\pi}} out of the integral.
Erf rule.
Fundamental theorem of calculus: plug in the limits.
Simplify.

Àwọn Àtòjọ-ẹ̀yàn

What is the difference between probability and statistics?

Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.

What does the law of large numbers promise?

That the average of many independent samples converges to the expected value. It says nothing about any single trial.

Diẹ̀ nínú Probability Theory

Sample spaces and the axioms Random variables and expectation The law of large numbers and the central limit theorem