Forest ErrAna StatDist - Revision history

Foretony: /* Mean of Discrete Probability Distribution */

2020-01-31T21:49:13Z

Mean of Discrete Probability Distribution

Foretony: /* Mean of Discrete Probability Distribution */

2020-01-31T21:48:31Z

Mean of Discrete Probability Distribution

Foretony: /* Mean of Discrete Probability Distribution */

2020-01-31T21:47:07Z

Mean of Discrete Probability Distribution

Foretony: /* Mean of Discrete Probability Distribution */

2020-01-31T21:45:36Z

Mean of Discrete Probability Distribution

Foretony: /* Biased and Unbiased variance */

2018-01-19T19:56:40Z

Biased and Unbiased variance

Foretony: /* discrete CDF example */

2017-02-10T19:55:19Z

discrete CDF example

Foretony: /* 2 SD rule of thumb for Gaussian PDF */

2017-02-10T19:46:20Z

2 SD rule of thumb for Gaussian PDF

Foretony: /* Integral Probability (Cumulative Distribution Function) */

2017-02-10T19:44:38Z

Integral Probability (Cumulative Distribution Function)

Foretony: /* Binomial with Large N becomes Gaussian */

2017-02-10T19:41:13Z

Binomial with Large N becomes Gaussian

Foretony: /* Binomial with Large N becomes Gaussian */

2017-02-10T19:40:08Z

Binomial with Large N becomes Gaussian

@@ Line 232: / Line 232: @@
 <math>x_i =</math> ith observable quantity
-<math>P(x_i) =</math> probability of observing <math>x_i</math> = Probability Mass Distribution for a discrete probability distribution
+<math>P(x_i) =</math> probability of observing <math>x_i</math> = Probability Distribution for a discrete probability distribution
 ====Mean of a continuous probability distibution====

@@ Line 222: / Line 222: @@
 For a Discrete probability distribution
-<math>\mu = E[x]=\lim_{N \rightarrow \infty} \frac{\sum_{i=1}^n x_i P(x_i)}{N}</math>
+<math>\mu = E[x]=\lim_{N \rightarrow \infty} \frac{\sum_{i=1}^n x_i N P(x_i)}{N}</math>
 where

@@ Line 222: / Line 222: @@
 For a Discrete probability distribution
-<math>\mu = E[x]=\lim_{N \rightarrow \infty} \frac{\sum_{i=1}^N x_i P(x_i)}{N}</math>
+<math>\mu = E[x]=\lim_{N \rightarrow \infty} \frac{\sum_{i=1}^n x_i P(x_i)}{N}</math>
 where

@@ Line 222: / Line 222: @@
 For a Discrete probability distribution
-<math>\mu = E[x]=\lim_{N \rightarrow \infty} \frac{\sum_{i=1}^n x_i P(x_i)}{N}</math>
+<math>\mu = E[x]=\lim_{N \rightarrow \infty} \frac{\sum_{i=1}^N x_i P(x_i)}{N}</math>
 where

@@ Line 208: / Line 208: @@
 :<math>\frac{\sum \sigma_i^2}{N} =</math> an average of all possible sample variance which should be equivalent to the "true" population variance.
-:<math>\Rightarrow \frac{\sum \sigma_i^2}{N} \sim \sum \left ( \frac{x_i-\bar{x}}{n} \right )^2</math> : if all the variances are the same this would be equivalent
+:<math>\Rightarrow \frac{\sum \sigma_i^2}{N} \sim \sum  \frac{(x_i-\bar{x})^2}{n}</math> : if all the variances are the same this would be equivalent
 : <math>\sigma^2 = \frac{n}{n-1}\frac{\sum(x_i-\bar{x})^2}{n}</math>

← Older revision		Revision as of 19:55, 10 February 2017
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−	There is an 8% probability that 5 or more ~~student~~ will fail the class	+	There is an 8% probability that 5 or more students will fail the class

	===== 2 SD rule of thumb for Gaussian PDF =====		===== 2 SD rule of thumb for Gaussian PDF =====

@@ Line 851: / Line 851: @@
-The above integral can only be done numerically by expanding the exponential in a power series
+The above integral can  be done numerically by expanding the exponential in a power series
 :<math>e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}</math>
@@ Line 861: / Line 861: @@
 : <math>=\left .  \frac{1}{\pi} \sum_{j=0}^{\infty} \frac{(-1)^j \left (\frac{x}{\sqrt{2}} \right)^{2j+1}}{j! (2j+1)} \right |_{x=\mu - 2\sigma}</math>
-No analytical for the probability but one which you can compute.
+The probability is one which you can compute.
 Below is a table representing the cumulative probability <math>P_G(x< \mu - \delta \mbox{ and }  x> \mu + \delta , \mu, \sigma)</math> for events to occur outside and interval of <math>\pm \delta</math> in a Gaussian distribution

← Older revision		Revision as of 19:44, 10 February 2017
Line 808:		Line 808:
	:<math>A_G(\Delta x, \mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}} \int_{\mu - \Delta x}^{\mu + \Delta x} e^{- \frac{1}{2} \left ( \frac{x - \mu}{\sigma}\right )^2} dx</math>		:<math>A_G(\Delta x, \mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}} \int_{\mu - \Delta x}^{\mu + \Delta x} e^{- \frac{1}{2} \left ( \frac{x - \mu}{\sigma}\right )^2} dx</math>

−	The advantage of this definition becomes apparent when you are ~~interesting~~ in quantifying the probability that a measurement would fall outside a range <math>\Delta x</math>.	+	The advantage of this definition becomes apparent when you are interested in quantifying the probability that a measurement would fall outside a range <math>\Delta x</math>.

← Older revision		Revision as of 19:41, 10 February 2017
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−	: <math>\ln \left [ n^{1/2} \right ] = \ln \left [ (n-s)^{1/2} \right ] = \ln \left [ (n+s)^{1/2} \right ]</math> For Large <math>n \ll s</math>	+	: <math>\ln \left [ n^{1/2} \right ] = \ln \left [ (n-s)^{1/2} \right ] = \ln \left [ (n+s)^{1/2} \right ]</math> For Large <math>n \gg s</math>

	:<math> \ln (R) = 2 n(\ln\left [ n \right ] -1 ) - (n-s)(\ln\left [ (n-s) \right ] -1 ) - (n+s)(\ln\left [ (n+s) \right ] -1 )</math>		:<math> \ln (R) = 2 n(\ln\left [ n \right ] -1 ) - (n-s)(\ln\left [ (n-s) \right ] -1 ) - (n+s)(\ln\left [ (n+s) \right ] -1 )</math>

@@ Line 678: / Line 678: @@
 similarly
-:<math>\ln\left [(n-s)! \right ] \sim   \ln \left [ \left (2 \pi \right)^{1/2} \right ] +\ln  \left [ (n-1)^{1/2} \right  ] + (n-s)(\ln\left [ (n-s) \right ] -1 )</math>
+:<math>\ln\left [(n-s)! \right ] \sim   \ln \left [ \left (2 \pi \right)^{1/2} \right ] +\ln  \left [ (n-s)^{1/2} \right  ] + (n-s)(\ln\left [ (n-s) \right ] -1 )</math>
-:<math>\ln\left [(n+s)! \right ] \sim   \ln \left [ \left (2 \pi \right)^{1/2} \right ] +\ln  \left [ (n+1)^{1/2} \right  ] + (n+s)(\ln\left [ (n+s) \right ] -1 )</math>
+:<math>\ln\left [(n+s)! \right ] \sim   \ln \left [ \left (2 \pi \right)^{1/2} \right ] +\ln  \left [ (n+s)^{1/2} \right  ] + (n+s)(\ln\left [ (n+s) \right ] -1 )</math>
 :<math>\Rightarrow \ln (R) = 2 \times  \left (\ln \left [ \left (2 \pi \right)^{1/2} \right ] +\ln  \left [ n^{1/2} \right  ] +n(\ln\left [ n \right ] -1 ) \right ) </math>
-:<math>- \left ( \ln \left [ \left (2 \pi \right)^{1/2} \right ] +\ln  \left [ (n-1)^{1/2} \right  ] + (n-s)(\ln\left [ (n-s) \right ] -1 )\right )</math>
+:<math>- \left ( \ln \left [ \left (2 \pi \right)^{1/2} \right ] +\ln  \left [ (n-s)^{1/2} \right  ] + (n-s)(\ln\left [ (n-s) \right ] -1 )\right )</math>
-:<math> -\left ( \ln \left [ \left (2 \pi \right)^{1/2} \right ] +\ln  \left [ (n+1)^{1/2} \right  ] + (n+s)(\ln\left [ (n+s) \right ] -1 )\right ) </math>
+:<math> -\left ( \ln \left [ \left (2 \pi \right)^{1/2} \right ] +\ln  \left [ (n+s)^{1/2} \right  ] + (n+s)(\ln\left [ (n+s) \right ] -1 )\right ) </math>
-: <math>= 2 \ln  \left [ n^{1/2} \right  ] +2 n(\ln\left [ n \right ] -1 )  - \ln  \left [ (n-1)^{1/2} \right  ] - (n-s)(\ln\left [ (n-s) \right ] -1 ) -\ln  \left [ (n+1)^{1/2} \right  ] - (n+s)(\ln\left [ (n+s) \right ] -1 )</math>
+: <math>= 2 \ln  \left [ n^{1/2} \right  ] +2 n(\ln\left [ n \right ] -1 )  - \ln  \left [ (n-s)^{1/2} \right  ] - (n-s)(\ln\left [ (n-s) \right ] -1 ) -\ln  \left [ (n+s)^{1/2} \right  ] - (n+s)(\ln\left [ (n+s) \right ] -1 )</math>
-: <math>\ln  \left [ n^{1/2} \right  ] = \ln  \left [ (n-1)^{1/2} \right  ]  = \ln  \left [ (n+1)^{1/2} \right  ]</math> For Large n
+: <math>\ln  \left [ n^{1/2} \right  ] = \ln  \left [ (n-s)^{1/2} \right  ]  = \ln  \left [ (n+s)^{1/2} \right  ]</math> For Large <math>n \ll s</math>
 :<math> \ln (R) =  2 n(\ln\left [ n \right ] -1 )   - (n-s)(\ln\left [ (n-s) \right ] -1 ) - (n+s)(\ln\left [ (n+s) \right ] -1 )</math>