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5.6: Formula Review

Last updated

Mar 17, 2025
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- 5.5: Chapter Review
- 5.7: Practice

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5.2 Properties of Continuous Probability Density Functions

Probability density function (pdf) $f(x)$ :

$f(x) \geq 0$
The total area under the curve $f(x)$ is one.

Cumulative distribution function (cdf): $P(X \leq x$ )

5.3 The Uniform Distribution

$X=$ a real number between $a$ and $b$ (in some instances, $X$ can take on the values $a$ and $b$ ). $a=$ smallest $X ; b=$ largest $X$

$X \sim U(\mathrm{a}, \mathrm{b})$

The mean is $\mu=\frac{a+b}{2}$

The standard deviation is $\sigma=\sqrt{\frac{(b-a)^2}{12}}$

Probability density function: $f(x)=\frac{1}{b-a}$ for $a \leq X \leq b$

Area to the Left of $x: P(X<x)=(x-a)\left(\frac{1}{b-a}\right)$

Area to the Right of $x: P(X>x)=(b-x)\left(\frac{1}{b-a}\right)$

Area Between $c$ and $d$ : $P(c<x<d)=\left(\right.$ base)(height) $=(d-c)\left(\frac{1}{b-a}\right)$

pdf: $f(x)=\frac{1}{b-a}$ for $a \leq x \leq b$
cdf: $P(X \leq x)=\frac{x-a}{b-a}$
mean $\mu=\frac{a+b}{2}$
standard deviation $\sigma=\sqrt{\frac{(b-a)^2}{12}}$
$P(c<X<d)=(d-c)\left(\frac{1}{b-a}\right)$

5.4 The Exponential Distribution

pdf: $f(x)=m e^{(-m x)}$ where $x \geq 0$ and $m>0$
cdf: $P(X \leq x)=1-e^{(-m x)}$
mean $\mu=\frac{1}{m}$
standard deviation $\sigma=\mu$
Additionally
- $P(X>x)=e^{(-m x)}$
- $P(a<X<b)=e^{(-m a)}-e^{(-m b)}$
Poisson probability: $P(X=x)=\frac{\mu^x e^{-\mu}}{x!}$ with mean and variance of $\mu$

5.2 Properties of Continuous Probability Density Functions

5.3 The Uniform Distribution

5.4 The Exponential Distribution

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