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User error: Socket error: Could not connect to "http://planetmath.org:8890/sparql?query=%0APREFIX+msc%3A+%3Chttp%3A%2F%2Fmsc2010.org%2Fresources%2FMSC%2F2010%2F%3E+PREFIX+skos%3A+%3Chttp%3A%2F%2Fwww.w3.org%2F2004%2F02%2Fskos%2Fcore%23%3E+PREFIX+dct%3A+%3Chttp%3A%2F%2Fpurl.org%2Fdc%2Fterms%2F%3E+PREFIX+local%3A+%3Chttp%3A%2F%2Flocal.virt%2F%3E+SELECT+%3Flabel+WHERE+%7B+GRAPH+%3Chttp%3A%2F%2Flocalhost%3A8890%2FDAV%2Fhome%2Fpm%2Frdf_sink%23this%3E+%7B+msc%3A11B25+skos%3AprefLabel+%3Flabel+FILTER+langMatches%28+lang%28%3Flabel%29%2C+%22en%22+%29+%7D%7D" (proxy: 0): Connection refused in ARC2_Reader in _sparql_request() (line 92 of /home/jcorneli/beta/sites/all/modules/sparql/sparql.module).
User error: missing stream in "getFormat" via ARC2_Reader in _sparql_request() (line 92 of /home/jcorneli/beta/sites/all/modules/sparql/sparql.module).
User error: missing stream in "readStream" http://planetmath.org:8890/sparql?query=%0APREFIX+msc%3A+%3Chttp%3A%2F%2Fmsc2010.org%2Fresources%2FMSC%2F2010%2F%3E+PREFIX+skos%3A+%3Chttp%3A%2F%2Fwww.w3.org%2F2004%2F02%2Fskos%2Fcore%23%3E+PREFIX+dct%3A+%3Chttp%3A%2F%2Fpurl.org%2Fdc%2Fterms%2F%3E+PREFIX+local%3A+%3Chttp%3A%2F%2Flocal.virt%2F%3E+SELECT+%3Flabel+WHERE+%7B+GRAPH+%3Chttp%3A%2F%2Flocalhost%3A8890%2FDAV%2Fhome%2Fpm%2Frdf_sink%23this%3E+%7B+msc%3A11B25+skos%3AprefLabel+%3Flabel+FILTER+langMatches%28+lang%28%3Flabel%29%2C+%22en%22+%29+%7D%7D via ARC2_Reader in _sparql_request() (line 92 of /home/jcorneli/beta/sites/all/modules/sparql/sparql.module).
User error: Socket error: Could not connect to "http://planetmath.org:8890/sparql?query=%0APREFIX+msc%3A+%3Chttp%3A%2F%2Fmsc2010.org%2Fresources%2FMSC%2F2010%2F%3E+PREFIX+skos%3A+%3Chttp%3A%2F%2Fwww.w3.org%2F2004%2F02%2Fskos%2Fcore%23%3E+PREFIX+dct%3A+%3Chttp%3A%2F%2Fpurl.org%2Fdc%2Fterms%2F%3E+PREFIX+local%3A+%3Chttp%3A%2F%2Flocal.virt%2F%3E+SELECT+%3Flabel+WHERE+%7B+GRAPH+%3Chttp%3A%2F%2Flocalhost%3A8890%2FDAV%2Fhome%2Fpm%2Frdf_sink%23this%3E+%7B+msc%3A00A06+skos%3AprefLabel+%3Flabel+FILTER+langMatches%28+lang%28%3Flabel%29%2C+%22en%22+%29+%7D%7D" (proxy: 0): Connection refused in ARC2_Reader in _sparql_request() (line 92 of /home/jcorneli/beta/sites/all/modules/sparql/sparql.module).
User error: missing stream in "getFormat" via ARC2_Reader in _sparql_request() (line 92 of /home/jcorneli/beta/sites/all/modules/sparql/sparql.module).
User error: missing stream in "readStream" http://planetmath.org:8890/sparql?query=%0APREFIX+msc%3A+%3Chttp%3A%2F%2Fmsc2010.org%2Fresources%2FMSC%2F2010%2F%3E+PREFIX+skos%3A+%3Chttp%3A%2F%2Fwww.w3.org%2F2004%2F02%2Fskos%2Fcore%23%3E+PREFIX+dct%3A+%3Chttp%3A%2F%2Fpurl.org%2Fdc%2Fterms%2F%3E+PREFIX+local%3A+%3Chttp%3A%2F%2Flocal.virt%2F%3E+SELECT+%3Flabel+WHERE+%7B+GRAPH+%3Chttp%3A%2F%2Flocalhost%3A8890%2FDAV%2Fhome%2Fpm%2Frdf_sink%23this%3E+%7B+msc%3A00A06+skos%3AprefLabel+%3Flabel+FILTER+langMatches%28+lang%28%3Flabel%29%2C+%22en%22+%29+%7D%7D via ARC2_Reader in _sparql_request() (line 92 of /home/jcorneli/beta/sites/all/modules/sparql/sparql.module).
User error: Socket error: Could not connect to "http://planetmath.org:8890/sparql?query=%0APREFIX+msc%3A+%3Chttp%3A%2F%2Fmsc2010.org%2Fresources%2FMSC%2F2010%2F%3E+PREFIX+skos%3A+%3Chttp%3A%2F%2Fwww.w3.org%2F2004%2F02%2Fskos%2Fcore%23%3E+PREFIX+dct%3A+%3Chttp%3A%2F%2Fpurl.org%2Fdc%2Fterms%2F%3E+PREFIX+local%3A+%3Chttp%3A%2F%2Flocal.virt%2F%3E+SELECT+%3Flabel+WHERE+%7B+GRAPH+%3Chttp%3A%2F%2Flocalhost%3A8890%2FDAV%2Fhome%2Fpm%2Frdf_sink%23this%3E+%7B+msc%3A00A05+skos%3AprefLabel+%3Flabel+FILTER+langMatches%28+lang%28%3Flabel%29%2C+%22en%22+%29+%7D%7D" (proxy: 0): Connection refused in ARC2_Reader in _sparql_request() (line 92 of /home/jcorneli/beta/sites/all/modules/sparql/sparql.module).
User error: missing stream in "getFormat" via ARC2_Reader in _sparql_request() (line 92 of /home/jcorneli/beta/sites/all/modules/sparql/sparql.module).
User error: missing stream in "readStream" http://planetmath.org:8890/sparql?query=%0APREFIX+msc%3A+%3Chttp%3A%2F%2Fmsc2010.org%2Fresources%2FMSC%2F2010%2F%3E+PREFIX+skos%3A+%3Chttp%3A%2F%2Fwww.w3.org%2F2004%2F02%2Fskos%2Fcore%23%3E+PREFIX+dct%3A+%3Chttp%3A%2F%2Fpurl.org%2Fdc%2Fterms%2F%3E+PREFIX+local%3A+%3Chttp%3A%2F%2Flocal.virt%2F%3E+SELECT+%3Flabel+WHERE+%7B+GRAPH+%3Chttp%3A%2F%2Flocalhost%3A8890%2FDAV%2Fhome%2Fpm%2Frdf_sink%23this%3E+%7B+msc%3A00A05+skos%3AprefLabel+%3Flabel+FILTER+langMatches%28+lang%28%3Flabel%29%2C+%22en%22+%29+%7D%7D via ARC2_Reader in _sparql_request() (line 92 of /home/jcorneli/beta/sites/all/modules/sparql/sparql.module).

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Given a probability distribution (density) function $f_{X}(x)$ with random variable $X$ and $x\in\mathbb{R}$ , a mode of $f_{X}(x)$ is a real number $\alpha$ such that:

1.

$f_{X}(\alpha)\neq\operatorname{min}(f_{X}(x))$ ,
2.

$f_{X}(\alpha)\geq f_{X}(z)$ for all $z\in\mathbb{R}$ .

The mode of $f_{X}$ is the set of all modes of $f_{X}$ (It is also customary to say denote the mode of $f_{X}$ to be elements within the mode of $f_{X}$ ). If the mode contains one element, then we say that $f_{X}$ is unimodal. If it has two elements, then $f_{X}$ is called bimodal. When $f_{X}$ has more than two modes, it is called multimodal.

if $\Omega=\{0,1,2,2,3,4,4,4,5,5,6,7,8\}$ is the sample space for the random variable $X$ , then the mode of the distribution function $f_{X}$ is 4.
if $\Omega=\{0,2,4,5,6,6,7,9,11,11,14,18\}$ is the sample space for $X$ , then the modes of $f_{X}$ are 6 and 11 and $f_{X}$ is bimodal.
For a binomial distribution with mean $np$ and variance $np(1-p)$ , the mode is

$\{\alpha\mid p(n+1)-1\leq\alpha\leq p(n+1)\}.$
For a Poisson distribution with integral sample space and mean $\lambda$ , if $\lambda$ is non-integral, then the mode is the largest integer less than or equal to $\lambda$ ; if $\lambda$ is an integer, then both $\lambda$ and $\lambda-1$ are modes.
For a normal distribution with mean $\mu$ and standard deviation $\sigma$ , the mode is $\mu$ .
For a gamma distribution with the shape parameter $\gamma$ , location parameter $\mu$ , and scale parameter $\beta$ , the mode is $\gamma-1$ if $\gamma>1$ .
Both the Pareto and the exponential distributions have mode = 0.

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