Continuous Random Variable: Difference between revisions

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{{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
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</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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) }

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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(c \leq x \leq d) = \int_c^d f(x) dx = F(d) - F(c) }


Total area under the curve must be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 } , as chances of events happening is 100% if the range includes all possible events.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^\infty f(x) dx = 1 }

There is no area under a single point

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(X = a) = 0 }

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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(X) = \mu_X = \int_{-\infty}^\infty x f(x) dx }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Var(X) = \sigma^2_X = \int_{-\infty}^\infty (x - \mu_X)^2 f(x) dx = \int_{-\infty}^\infty x^2 f(x) dx - \mu_X^2 }

Median and Percentile

The a-th percentileis the point at which a percent the area under the curve is to one side. You want Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(X \leq x) } 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim Uniform(a, b) }

Uniform random variable is described by two parameters: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } is minimum, and is maximum. It has a rectangular distribution, where every point has the same probability density.

PDF

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \begin{cases} \frac{ 1 }{ b - a } & a \leq x \leq b \\ 0 & \text{otherwise} \end{cases} }

CDF

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \begin{cases} 0 & x < a \\ \frac{ x - a }{ b - a } & a \leq x \leq b \\ 1 & x > b \end{cases} }

Mean

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_X = \frac{ a + b }{ 2 } }

Variance

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ 1 }{ 12 } (b - a)^2 }

Exponential Distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim Exp(\lambda)}

The exponential distribution models the time ntil a particular event occurs.

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