5.6: Formula Review
- Page ID
- 45938
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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 \(\mu\)