Pattern Recognition - Chapter 2: Normal distribution
1. Normal distribution
![](https://static.wixstatic.com/media/64c52b_10fb86051eb34887ad020e9085e198d0~mv2.png/v1/fill/w_265,h_90,al_c,q_85,enc_auto/64c52b_10fb86051eb34887ad020e9085e198d0~mv2.png)
![](https://static.wixstatic.com/media/64c52b_b228cd4d57c745e286840c173bca56c6~mv2.png/v1/fill/w_769,h_558,al_c,q_90,enc_auto/64c52b_b228cd4d57c745e286840c173bca56c6~mv2.png)
(*Source: https://en.wikipedia.org/wiki/Normal_distribution)
![](https://static.wixstatic.com/media/64c52b_8dc81da49f2f490a9ca2dbc328f22046~mv2.png/v1/fill/w_265,h_90,al_c,q_85,enc_auto/64c52b_8dc81da49f2f490a9ca2dbc328f22046~mv2.png)
2. Multivariate normal densities
![](https://static.wixstatic.com/media/64c52b_83eff63d9a8f4c45895075550cc04309~mv2.png/v1/fill/w_295,h_71,al_c,q_85,enc_auto/64c52b_83eff63d9a8f4c45895075550cc04309~mv2.png)
![](https://static.wixstatic.com/media/64c52b_42d8597cabee4f43af8b1d20265883d5~mv2.png/v1/fill/w_191,h_185,al_c,q_85,enc_auto/64c52b_42d8597cabee4f43af8b1d20265883d5~mv2.png)
![](https://static.wixstatic.com/media/64c52b_69205dedea5f4da5b7c2ce2628fb68bd~mv2.png/v1/fill/w_320,h_238,al_c,q_85,enc_auto/64c52b_69205dedea5f4da5b7c2ce2628fb68bd~mv2.png)
(*Source: http://ballistipedia.com/index.php?title=Closed_Form_Precision)
![](https://static.wixstatic.com/media/64c52b_7fb62ee1d50b45a69a907aacf21e96ea~mv2.png/v1/fill/w_197,h_186,al_c,q_85,enc_auto/64c52b_7fb62ee1d50b45a69a907aacf21e96ea~mv2.png)
![](https://static.wixstatic.com/media/64c52b_3f2da60dd7ef4beaab5fb4f7f691d9bb~mv2.png/v1/fill/w_423,h_140,al_c,q_85,enc_auto/64c52b_3f2da60dd7ef4beaab5fb4f7f691d9bb~mv2.png)
![](https://static.wixstatic.com/media/64c52b_28bdbf6558794c09808d78d66fd9ee94~mv2.png/v1/fill/w_233,h_233,al_c,q_85,enc_auto/64c52b_28bdbf6558794c09808d78d66fd9ee94~mv2.png)
![](https://static.wixstatic.com/media/64c52b_b531066ce9be4ca5922eb09bf203cf21~mv2.png/v1/fill/w_305,h_271,al_c,q_85,enc_auto/64c52b_b531066ce9be4ca5922eb09bf203cf21~mv2.png)
(*Source: https://blogs.sas.com/content/iml/files/2012/07/mvnormalpdf.png)
3. Mixture densities
![](https://static.wixstatic.com/media/64c52b_988891b7ed2641ce81a6fadd733d8191~mv2.png/v1/fill/w_290,h_256,al_c,q_85,enc_auto/64c52b_988891b7ed2641ce81a6fadd733d8191~mv2.png)
![](https://static.wixstatic.com/media/64c52b_a4038db984cb4e07a1180b7361db25e9~mv2.png/v1/fill/w_255,h_63,al_c,q_85,enc_auto/64c52b_a4038db984cb4e07a1180b7361db25e9~mv2.png)
![](https://static.wixstatic.com/media/64c52b_a8899faa41cf4b94b33f98f849fd5e42~mv2.png/v1/fill/w_421,h_70,al_c,q_85,enc_auto/64c52b_a8899faa41cf4b94b33f98f849fd5e42~mv2.png)
4. The product of Gaussian distributions
4.1. The product of two uni-variate Gaussian distribution
![](https://static.wixstatic.com/media/64c52b_dbf6eb9d766d4d3483fd0cfe7dbb1a70~mv2.png/v1/fill/w_238,h_78,al_c,q_85,enc_auto/64c52b_dbf6eb9d766d4d3483fd0cfe7dbb1a70~mv2.png)
![](https://static.wixstatic.com/media/64c52b_69232e1f16994e74a1df2beaa5ed3640~mv2.png/v1/fill/w_231,h_71,al_c,q_85,enc_auto/64c52b_69232e1f16994e74a1df2beaa5ed3640~mv2.png)
![](https://static.wixstatic.com/media/64c52b_c9eedbb68fd24f90a7075d4ee528115e~mv2.png/v1/fill/w_320,h_74,al_c,q_85,enc_auto/64c52b_c9eedbb68fd24f90a7075d4ee528115e~mv2.png)
Equation (1)
![](https://static.wixstatic.com/media/64c52b_2d8079dc8284486fb710f44a7dfd7cdf~mv2.png/v1/fill/w_456,h_75,al_c,q_85,enc_auto/64c52b_2d8079dc8284486fb710f44a7dfd7cdf~mv2.png)
![](https://static.wixstatic.com/media/64c52b_1aa1edd58ddb4d70b4987142e011165b~mv2.png/v1/fill/w_320,h_47,al_c,q_85,enc_auto/64c52b_1aa1edd58ddb4d70b4987142e011165b~mv2.png)
Equation (2)
![](https://static.wixstatic.com/media/64c52b_7bcdfd0311214b718460c335bfe30a16~mv2.png/v1/fill/w_297,h_96,al_c,q_85,enc_auto/64c52b_7bcdfd0311214b718460c335bfe30a16~mv2.png)
![](https://static.wixstatic.com/media/64c52b_c18b1ff0537d479cb25331f029f02b19~mv2.png/v1/fill/w_297,h_207,al_c,q_85,enc_auto/64c52b_c18b1ff0537d479cb25331f029f02b19~mv2.png)
![](https://static.wixstatic.com/media/64c52b_e7ec970a71a74d569c50dd9da9c6e664~mv2.png/v1/fill/w_169,h_85,al_c,q_85,enc_auto/64c52b_e7ec970a71a74d569c50dd9da9c6e664~mv2.png)
![](https://static.wixstatic.com/media/64c52b_20f3c87a342149dd895754cf34f6d7af~mv2.png/v1/fill/w_191,h_70,al_c,q_85,enc_auto/64c52b_20f3c87a342149dd895754cf34f6d7af~mv2.png)
![](https://static.wixstatic.com/media/64c52b_ee2092235bc04540bd65f0ca9925e244~mv2.png/v1/fill/w_502,h_139,al_c,q_85,enc_auto/64c52b_ee2092235bc04540bd65f0ca9925e244~mv2.png)
![](https://static.wixstatic.com/media/64c52b_fb281153c01f40eab8028320a63ac58b~mv2.png/v1/fill/w_352,h_80,al_c,q_85,enc_auto/64c52b_fb281153c01f40eab8028320a63ac58b~mv2.png)
![](https://static.wixstatic.com/media/64c52b_634bc339699646d68cbb21640c63a06e~mv2.png/v1/fill/w_622,h_108,al_c,q_85,enc_auto/64c52b_634bc339699646d68cbb21640c63a06e~mv2.png)
![](https://static.wixstatic.com/media/64c52b_87201d766f454e8a9094f9f2191f00df~mv2.png/v1/fill/w_579,h_104,al_c,q_85,enc_auto/64c52b_87201d766f454e8a9094f9f2191f00df~mv2.png)
![](https://static.wixstatic.com/media/64c52b_c929129fc32649c6809d1c6b451cf6e8~mv2.png/v1/fill/w_400,h_58,al_c,q_85,enc_auto/64c52b_c929129fc32649c6809d1c6b451cf6e8~mv2.png)
Equation (3)
![](https://static.wixstatic.com/media/64c52b_4d345456f72049959411c73561a8b151~mv2.png/v1/fill/w_172,h_89,al_c,q_85,enc_auto/64c52b_4d345456f72049959411c73561a8b151~mv2.png)
![](https://static.wixstatic.com/media/64c52b_9cad95fb60fb421e8f47202cae7a7a3a~mv2.png/v1/fill/w_220,h_50,al_c,q_85,enc_auto/64c52b_9cad95fb60fb421e8f47202cae7a7a3a~mv2.png)
![](https://static.wixstatic.com/media/64c52b_be073bd9eba640e0bc9c4ff58dcbaa4d~mv2.png/v1/fill/w_601,h_86,al_c,q_85,enc_auto/64c52b_be073bd9eba640e0bc9c4ff58dcbaa4d~mv2.png)
![](https://static.wixstatic.com/media/64c52b_3f37893cb789439595fb566fe29f2977~mv2.png/v1/fill/w_320,h_70,al_c,q_85,enc_auto/64c52b_3f37893cb789439595fb566fe29f2977~mv2.png)
![](https://static.wixstatic.com/media/64c52b_ad624981e9f9467faa0137b7e298be2a~mv2.png/v1/fill/w_225,h_79,al_c,q_85,enc_auto/64c52b_ad624981e9f9467faa0137b7e298be2a~mv2.png)
![](https://static.wixstatic.com/media/64c52b_ca5f95fded3545d99dbfde6f72fa5235~mv2.png/v1/fill/w_158,h_78,al_c,q_85,enc_auto/64c52b_ca5f95fded3545d99dbfde6f72fa5235~mv2.png)
![](https://static.wixstatic.com/media/64c52b_cbd96f45a1f5454e9a6dc48b0c9f98e7~mv2.png/v1/fill/w_320,h_87,al_c,q_85,enc_auto/64c52b_cbd96f45a1f5454e9a6dc48b0c9f98e7~mv2.png)
4.2. The product of n uni-variate Gaussian distributions
![](https://static.wixstatic.com/media/64c52b_41155370a33b464292f5c5c526fcda65~mv2.png/v1/fill/w_400,h_122,al_c,q_85,enc_auto/64c52b_41155370a33b464292f5c5c526fcda65~mv2.png)
![](https://static.wixstatic.com/media/64c52b_a7097647d7334ad38431ea611dcfb7e0~mv2.png/v1/fill/w_640,h_120,al_c,q_85,enc_auto/64c52b_a7097647d7334ad38431ea611dcfb7e0~mv2.png)
![](https://static.wixstatic.com/media/64c52b_b3241e95d34a456083175989ecf30e75~mv2.png/v1/fill/w_640,h_70,al_c,q_85,enc_auto/64c52b_b3241e95d34a456083175989ecf30e75~mv2.png)