|
|
Line 1: |
Line 1: |
| {{Probability distribution
| |
| | name = Exponential
| |
| | type = continuous
| |
| | pdf_image = [[File:Exponential_distribution_pdf_-_public_domain.svg|325px|plot of the probability density function of the exponential distribution]]
| |
| | cdf_image = [[File:Exponential_distribution_cdf_-_public_domain.svg|325px|Cumulative distribution function]]
| |
| | parameters = <math>\lambda > 0,</math> rate, or inverse [[scale parameter|scale]]
| |
| | support = <math>x \in [0, \infty)</math>
| |
| | pdf = <math>\lambda e^{-\lambda x}</math>
| |
| | cdf = <math>1 - e^{-\lambda x}</math>
| |
| | quantile = <math>-\frac{\ln(1 - p)}{\lambda}</math>
| |
| | mean = <math>\frac{1}{\lambda}</math>
| |
| | median = <math>\frac{\ln 2}{\lambda}</math>
| |
| | mode = <math>0</math>
| |
| | variance = <math> \frac{1}{\lambda^2}</math>
| |
| | skewness = <math>2</math>
| |
| | kurtosis = <math>6</math>
| |
| | entropy = <math>1 - \ln\lambda</math>
| |
| | mgf = <math>\frac{\lambda}{\lambda-t}, \text{ for } t < \lambda</math>
| |
| | char = <math>\frac{\lambda}{\lambda-it}</math>
| |
| | fisher = <math>\frac{1}{\lambda^2}</math>
| |
| | KLDiv = <math>\ln\frac{\lambda_0}{\lambda} + \frac{\lambda}{\lambda_0} - 1</math>
| |
| |ES=<math>-\frac{\ln(1 - p) + 1}{\lambda}</math>|bPOE=<math>e^{1-\lambda x}</math>}}
| |
|
| |
|
| |
| Continuous random variables have an inifinite number of values for any | | Continuous random variables have an inifinite number of values for any |
| given interval. While similar, the approach to analysis is very | | given interval. While similar, the approach to analysis is very |
Line 114: |
Line 90: |
| </math> | | </math> |
|
| |
|
| = Exponential Distribution <math> X \sim Exp(\lambda) </math> = | | = Exponential Distribution <math>X \sim Exp(\lambda)</math> = |
|
| |
|
| The exponential distribution models the time ntil a particular event | | The exponential distribution models the time ntil a particular event |
| occurs. | | occurs. |
Revision as of 08:25, 1 March 2024
Continuous random variables have an inifinite number of values for any
given interval. While similar, the approach to analysis is very
different from discrete variables
- Summation becomes integration
- Probability becomes area under a curve
Probability Distribution Function
The probability density function (pdf) maps a continuous variable to a
probability density.
As the name "density" suggests, the area under the pdf curve between a
range is the probability of the variable being in that range.
Total area under the curve must be , as chances of
events happening is 100% if the range includes all possible events.
There is no area under a single point
Mean and Variance
The mean and variance calculations are pretty much the same as that of
discrete random variables, except the summations are swapped out for
integrals.
Median and Percentile
The a-th percentileis the point at which a percent the area under the
curve is to one side. You want to be a%, the
calculation of which is in the page above.
By the same logic, the quartiles are at 25%, 50%, and 75% accordingly.
Uniform Distribution
Uniform random variable is described by two parameters:
is minimum, and is maximum. It has a rectangular
distribution, where every point has the same probability density.
PDF
CDF
Mean
Variance
Exponential Distribution
The exponential distribution models the time ntil a particular event
occurs.