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In mathematics a Gaussian Distribution Function is a function of the form:

In mathematics a Gaussian Distribution Function is a function of the form:

Distribution

Functional Form

Mean

Standard Deviation

Ganussian

    fg(x)= 1 2π σ 2 e (xa) 2 2 σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qadaWcaaqaaiaaigdaaeaadaGcaaqaaiaaikdacqaHapaCcqaHdpWC daqhaaWcbaaabaGaaGOmaaaaaeqaaaaakiaadwgadaahaaWcbeqaam aalaaabaGaeyOeI0IaaiikaiaadIhacqGHsislcaWGHbGaaiykamaa CaaameqabaGaaGOmaaaaaSqaaiaaikdacqaHdpWCdaqhaaadbaaaba GaaGOmaaaaaaaaaaaa@497D@       

a

σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacqaHdpWCdaWgaaWceaqabeaaaeaaaaqabaaaaa@3ACE@

The Gaussian distribution is one of the most commonly used probability distribution for applications. If the number of events is very large, then the Gaussian distribution function may be used to describe physical events. The Gaussian distribution is a continuous function which approximates the exact binomial distribution of events :

P(y)= 1 σ 2π e (yμ) 2 σ 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacaWGqbGaaiikaiaacMhacaGGPaGaeyypa0ZaaSaaaeaacaaIXaaa baGaeq4Wdm3aaOaaaeaacaaIYaGaeqiWdahaleqaaaaakiaadwgada WcaaqaaiabgkHiTiaacIcacaWG5bGaeyOeI0IaeqiVd0Maaiykaaqa aiaaikdacqaHdpWCdaahaaWcbeqaaiaaikdaaaaaaOWaaWbaaSqabe aacaaIYaaaaaaa@4D71@

μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacqaH8oqBaaa@3A8E@ :mean of distribution   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F6@    σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacqaHdpWCdaahaaWcbeqaaiaaikdaaaaaaa@3B83@ :variance of the distribution    y is a continuous variable (y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacaGGOaGaeyOeI0IaeyOhIuQaeyizImQaamyEaiabgsMiJkabg6Hi LkaacMcaaaa@4267@   

Probability (P) of y being in the range [a, b] is given by an integral

  P(y<a<b)= 1 σ 2π a b e (yμ) 2 2 σ 2 d y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacaWGqbGaaiikaiaacMhacqGH8aapcaGGHbGaeyipaWJaaiOyaiaa cMcacqGH9aqpdaWcaaqaaiaaigdaaeaacqaHdpWCdaGcaaqaaiaaik dacqaHapaCaSqabaaaaOWaa8qCaeaacaWGLbWaaWbaaSqabeaadaWc aaqaaiabgkHiTiaacIcacaWG5bGaeyOeI0IaeqiVd0MaaiykamaaCa aameqabaGaaGOmaaaaaSqaaiaaikdacqaHdpWCdaahaaadbeqaaiaa ikdaaaaaaaaaaSqaaiaadggaaeaacaWGIbaaniabgUIiYdGccaWGKb WaaSbaaSqaaiaadMhaaeqaaaaa@57C7@

Total area under the curve is normalized to one.

The probability integral: P(<y<)= 1 σ 2π e (yμ) 2 2 σ 2 d y =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacaWGqbGaaiikaiabgkHiTiabg6HiLkabgYda8iaadMhacqGH8aap cqGHEisPcaGGPaGaeyypa0ZaaSaaaeaacaaIXaaabaGaeq4Wdm3aaO aaaeaacaaIYaGaeqiWdahaleqaaaaakmaapehabaGaamyzamaaCaaa leqabaWaaSaaaeaacqGHsislcaGGOaGaamyEaiabgkHiTiabeY7aTj aacMcadaahaaadbeqaaiaaikdaaaaaleaacaaIYaGaeq4Wdm3aaWba aWqabeaacaaIYaaaaaaaaaaaleaacqGHsislcqGHEisPaeaacqGHEi sPa0Gaey4kIipakiaadsgadaWgaaWcbaGaamyEaaqabaGccqGH9aqp caaIXaaaaa@5D98@

The Gaussian distribution is also commonly called the "normal distribution" and is often described as a "bell-shaped curve".

The Diagram shows Normalized Gaussian curves with expected value μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacqaH8oqBdaWgaaWcbaaabeaaaaa@3ABA@  and variance σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacqaHdpWCdaqhaaWcbaaabaGaaGOmaaaaaaa@3B84@  The corresponding parameters are a= 1 (σ (2π)) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qadaWcaaqaaiaaigdaaeaacaGGOaGaeq4Wdm3aaOaaaeaacaGGOaGa aGOmaiabec8aWjaacMcacaGGPaaaleqaaaaaaaa@40AC@ , b= μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacqaH8oqBaaa@3A8E@ , c= σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacqaHdpWCaaa@3A9B@ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F6@ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F6@

The normal distribution with mean 0 and standard deviation 1 N(0;1)is called the standard normal distribution.

A random variable with the standard normal distribution is called a standard normal random variable and is usually denoted by Z.

The cumulative probability distribution of the standard normal distribution P(Zz) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacaWGqbGaaiikaiaadQfacqGHKjYOcaWG6bGaaiykaaaa@3E98@

has been tabulated and is used to calculate probabilities for any normal random variable.

Smiley face

P(Z z 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacaWGqbGaaiikaiaadQfacqGHKjYOcaWG6bWaaSbaaSqaaiaaicda aeqaaOGaaiykaaaa@3F88@  gives the area under the curve to the left of z 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacaWG6bWaaSbaaSqaaiaaicdaaeqaaaaa@3ABC@

P( z 0 Z z 1 )=P(Z z 1 )P(Z z 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacaWGqbGaaiikaiaadQhadaWgaaWcbaGaaGimaaqabaGccqGHKjYO caWGAbGaeyizImQaamOEamaaBaaaleaacaaIXaaabeaakiaacMcacq GH9aqpcaWGqbGaaiikaiaadQfacqGHKjYOcaWG6bWaaSbaaSqaaiaa igdaaeqaaOGaaiykaiabgkHiTiaadcfacaGGOaGaamOwaiabgsMiJk aadQhadaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa@5283@  

The distribution is symmetric P(Z z 0 )=P(Z z 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaG+aOpa6d8 qacaWGqbGaaiikaiaadQfacqGHKjYOcaWG6bWaaSbaaSqaaiaaicda aeqaaOGaaiykaiabg2da9iaaccfacaGGOaGaaiOwaiabgwMiZkabgk HiTiaacQhadaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa@483A@

Example: Look at the graph below and suppose Z is a standard random variable. Calculate:

  P(Z1.1) P(Z>0.8) P(Z1.52) P(0.4Z1.32) P(0.2Z0.34) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaa6dG +aOpWdbiaadcfacaGGOaGaamOwaiabgsMiJkaaigdacaGGUaGaaGym aiaacMcaaeaacaWGqbGaaiikaiaadQfacqGH+aGpcaaIWaGaaiOlai aaiIdacaGGPaaabaGaamiuaiaacIcacaWGAbGaeyizImQaeyOeI0Ia aGymaiaac6cacaaI1aGaaGOmaiaacMcaaeaacaWGqbGaaiikaiaaic dacaGGUaGaaGinaiabgsMiJkaadQfacqGHKjYOcaaIXaGaaiOlaiaa iodacaaIYaGaaiykaaqaaiaadcfacaGGOaGaeyOeI0IaaGimaiaac6 cacaaIYaGaeyizImQaamOwaiabgsMiJkaaicdacaGGUaGaaG4maiaa isdacaGGPaaaaaa@669F@

 



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