Calculus Small Changes, Big Ideas

Section 9.4 Probability distributions

• Brief review of probability

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• A function \(f\) is a probability density function if

  \begin{equation*} \int_{-\infty}^{\infty} f(x)\dd{x}=1\text{.} \end{equation*}

  Then we say the probability that \(a\le x\le b\) is

  \begin{equation*} \Prob(a\le x\le b)=\int_{a}^{b} f(x)\dd{x}\text{.} \end{equation*}

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• Mean of a random variable \(X\text{:}\)

  \begin{equation*} \mu=\EV(X)=\int_{-\infty}^{\infty} xf(x)\dd{x} \end{equation*}

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• Variance:

  \begin{equation*} \Var(X)=\EV[(X-\EV(X))^{2}]=\EV(X^{2})-\EV(X)^{2} \end{equation*}

  Standard deviation:

  \begin{equation*} \sigma(X)=\sqrt{\Var(X)} \end{equation*}

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• Gaussian integral:

  \begin{equation*} \int_{-\infty}^{\infty}\ee^{-x^2}\dd{x}=\sqrt{\pii} \end{equation*}

  Error function:

  \begin{equation*} \erf(x)=\frac{2}{\sqrt{\pii}}\int_{0}^{x}\ee^{-x^2}\dd{x} \end{equation*}

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