{"id":2293,"date":"2017-06-15T22:02:29","date_gmt":"2017-06-15T22:02:29","guid":{"rendered":"http:\/\/10.10.10.4:9191\/softlect\/?p=2293"},"modified":"2019-06-22T15:46:25","modified_gmt":"2019-06-22T15:46:25","slug":"normal-and-probability-distribution","status":"publish","type":"post","link":"http:\/\/softlect.com\/index.php\/normal-and-probability-distribution\/","title":{"rendered":"Normal and Probability Distribution"},"content":{"rendered":"

Normal distribution is a continuous probability distribution. It is also called Gaussian distribution.
\nThe normal distribution density function f<\/em>(z) is called the Bell Curve because it has the shape that resembles a bell.
\nStandard normal distribution table is used to find the area under the f<\/em>(z<\/em>) function in order to find the probability of a specified range of distribution.<\/p>\n

Normal distribution function<\/p>\n

When random variable X has normal distribution,<\/p>\n

\"\"
\n\"\"
\n\"\"<\/p>\n

The probability density function and cumulative distribution function of the normal distribution:<\/p>\n

Probability density function (pdf)<\/h4>\n

The probability density function is given by:<\/p>\n

<\/p>\n

X is the random variable.
\n\u03bc is the mean value.
\n\u03c3 is the standard deviation (std) value.
\ne = 2.7182818… constant.
\n\u03c0 = 3.1415926… constant.<\/p>\n

Cumulative distribution function<\/h4>\n

The cumulative distribution function is given by:<\/p>\n

\"F_{X}(x)=\\frac{1}{\\sigma\\sqrt{2\\pi}}\\int_{-\\infty}^{x}e^{-\\frac{(y-\\mu)^2}{2\\sigma^2}}dy\"<\/p>\n

X is the random variable.<\/p>\n

\u03bc is the mean value.<\/p>\n

\u03c3 is the standard deviation (std) value.<\/p>\n

e = 2.7182818… constant.<\/p>\n

\u03c0 = 3.1415926… constant.<\/p>\n

Standard normal distribution function<\/p>\n

When \"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Then the probability density function and cumulative distribution function of the standard normal distribution:<\/p>\n

Probability density function<\/h4>\n

 <\/p>\n

\"\"<\/p>\n

Cumulative distribution function<\/h4>\n

 <\/p>\n

\"\"<\/p>\n

 <\/p>\n

In probability and statistics distribution<\/em> is a characteristic of a random variable, describes the probability of the random variable in each value.<\/p>\n

Each distribution has a certain probability density function and probability distribution function.<\/p>\n

Though there are indefinite number of probability distributions, there are several common distributions in use.<\/p>\n

Cumulative distribution function<\/p>\n

The probability distribution is described by the cumulative distribution function F(x),<\/p>\n

which is the probability of random variable X to get value smaller than or equal to x:<\/p>\n

F<\/em>(x<\/em>) = P<\/em>(X<\/em> \u2264 x<\/em>)<\/p>\n

The cumulative distribution function F(x) is calculated by integration of the probability density function f(u) of continuous random variable X.<\/p>\n

\"\"<\/p>\n

The cumulative distribution function F(x) is calculated by summation of the probability mass function P(u) of discrete random variable X.<\/p>\n

\"\"<\/p>\n

Continuous distribution is the distribution of a continuous random variable<\/p>\n

 <\/p>\n

Continuous distributions table<\/h4>\n
\n
\n
Distribution name<\/strong><\/div>\n
Distribution symbol<\/strong><\/div>\n
Probability density function (pdf)<\/strong><\/div>\n
Mean<\/strong><\/div>\n
Variance<\/strong><\/div>\n<\/div>\n
\n
<\/div>\n
<\/div>\n
<\/p>\n

f<\/em>X<\/em><\/sub>(x<\/em>)<\/p>\n

<\/strong><\/div>\n

<\/p>\n

\u03bc<\/em> = E<\/em>(X<\/em>)<\/p>\n

<\/strong><\/div>\n

<\/p>\n

\u03c3<\/em>2<\/sup> = Var<\/em>(X<\/em>)<\/p>\n

<\/strong><\/div>\n<\/div>\n

\n
Normal \/ gaussian<\/div>\n
\n

X <\/em>~ N<\/em>(\u03bc,\u03c32<\/sup>)<\/p>\n<\/div>\n

<\/div>\n
\u03bc<\/em><\/div>\n
\u03c3<\/em>2<\/sup><\/div>\n<\/div>\n
\n
Uniform<\/div>\n
\n

X <\/em>~ U<\/em>(a<\/em>,b<\/em>)<\/p>\n<\/div>\n

<\/div>\n
<\/div>\n
<\/div>\n<\/div>\n
\n
Exponential<\/div>\n
X <\/em>~ exp<\/em>(\u03bb)<\/div>\n
<\/div>\n
<\/div>\n
<\/div>\n<\/div>\n
\n
Gamma<\/div>\n
X <\/em>~ gamma<\/em>(c<\/em>, \u03bb)<\/div>\n
<\/p>\n

x<\/em> > 0, c<\/em> > 0, \u03bb > 0<\/p>\n<\/div>\n

<\/div>\n
<\/div>\n<\/div>\n
\n
Chi square<\/div>\n
\n

X <\/em>~\u03c7 2<\/sup>(k<\/em>)<\/p>\n<\/div>\n

<\/div>\n
\n

k<\/em><\/p>\n<\/div>\n

\n

2k<\/em><\/p>\n<\/div>\n<\/div>\n

\n
F<\/div>\n
\n

X <\/em>~ F <\/em>(k<\/em>1<\/sub>, k2<\/sub><\/em>)<\/p>\n<\/div>\n

<\/div>\n
<\/div>\n
<\/div>\n<\/div>\n
\n
Log-normal<\/div>\n
\n

X <\/em>~ LN<\/em>(\u03bc,\u03c32<\/sup>)<\/p>\n<\/div>\n

<\/div>\n
<\/div>\n
<\/div>\n<\/div>\n<\/div>\n

Discrete distribution is the distribution of a discrete random variable.<\/p>\n

 <\/p>\n

Discrete distributions table<\/h4>\n
\n
\n
Distribution name<\/strong><\/div>\n
Distribution symbol<\/strong><\/div>\n
Probability mass function (pmf)<\/strong><\/div>\n
Mean<\/strong><\/div>\n
Variance<\/strong><\/div>\n<\/div>\n
\n
<\/div>\n
<\/div>\n
<\/p>\n

fx<\/sub><\/em>(k<\/em>) = P<\/em>(X<\/em>=k<\/em>)<\/p>\n

k<\/em> = 0,1,2,…<\/p>\n

<\/strong><\/div>\n

E<\/em>(x<\/em>)<\/strong><\/div>\n
Var<\/em>(x<\/em>)<\/strong><\/div>\n<\/div>\n
\n
Binomial<\/div>\n
X <\/em>~ Bin<\/em>(n<\/em>,p<\/em>)<\/div>\n
<\/div>\n
np<\/em><\/div>\n
np<\/em>(1-p<\/em>)<\/div>\n<\/div>\n
\n
Poisson<\/div>\n
X <\/em>~ Poisson<\/em>(\u03bb)<\/div>\n
\"\"<\/p>\n

\u03bb \u2265 0<\/p>\n<\/div>\n

\u03bb<\/div>\n
\u03bb<\/div>\n<\/div>\n
\n
Uniform<\/div>\n
X <\/em>~ U<\/em>(a,b<\/em>)\n<\/div>\n
<\/div>\n
<\/div>\n
<\/div>\n<\/div>\n
\n
Geometric<\/div>\n
X <\/em>~ Geom<\/em>(p<\/em>)<\/div>\n
<\/div>\n
<\/div>\n
<\/div>\n<\/div>\n
\n
Hyper-geometric<\/div>\n
X <\/em>~ HG<\/em>(N<\/em>,K<\/em>,n<\/em>)<\/div>\n
\"\"<\/p>\n

N <\/em>= 0,1,2,…<\/p>\n

K <\/em>= 0,1,..,N<\/em><\/p>\n

n <\/em>= 0,1,…,N<\/em><\/p>\n<\/div>\n

<\/div>\n
<\/div>\n<\/div>\n
\n
Bernoulli<\/div>\n
X <\/em>~ Bern<\/em>(p<\/em>)<\/div>\n
<\/div>\n
p<\/em><\/div>\n
p<\/em>(1-p<\/em>)<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Normal distribution is a continuous probability distribution. It is also called Gaussian distribution. The normal distribution density function f(z) is called the Bell Curve because it has the shape that… <\/p>\n","protected":false},"author":1,"featured_media":3122,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[34],"tags":[],"aioseo_notices":[],"amp_enabled":true,"_links":{"self":[{"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/posts\/2293"}],"collection":[{"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/comments?post=2293"}],"version-history":[{"count":21,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/posts\/2293\/revisions"}],"predecessor-version":[{"id":3562,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/posts\/2293\/revisions\/3562"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/media\/3122"}],"wp:attachment":[{"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/media?parent=2293"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/categories?post=2293"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/tags?post=2293"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}