maths.free › Probability Theory › The common distributions

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.

작업 예제 · 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

단계별로

  1. \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).

  2. \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.

  3. \int e^{- \frac{x^{2}}{2}}\, dx = \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}

    Erf rule.

  4. 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.

  5. = \operatorname{erf}{\left(0.98 \sqrt{2} \right)} \approx 0.95000

    Simplify.

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

이제 너 문제를 선택하거나, 타이핑하거나, 자신만의 문제를 그려보세요. 모든 단계는 그림으로 표시되며, 답은 질문을 할 때까지 숨겨져 있습니다.

자신의 작업을 유지

무료 계정은 모든 수업에 노트를 추가, 당신이 완료 한 기록, 당신이 해결 한 문제를 한 곳에, 그리고 당신이이 페이지에 대해 물어 수업 지도자. 수학 자체는 모두에게 열려 있습니다, 로그인 여부.

가입하기 로그인

여기서 사용된 기호

기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.

어떻게: The common distributions

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

사람들이 묻는 질문

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.

에 더 Probability Theory

Sample spaces and the axiomsRandom variables and expectationThe law of large numbers and the central limit theorem