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<section id="numpy-random-randomstate-standard-t">
<h1>numpy.random.RandomState.standard_t<a class="headerlink" href="#numpy-random-randomstate-standard-t" title="Link to this heading">#</a></h1>
<p>method</p>
<dl class="py method">
<dt class="sig sig-object py" id="numpy.random.RandomState.standard_t">
<span class="sig-prename descclassname"><span class="pre">random.RandomState.</span></span><span class="sig-name descname"><span class="pre">standard_t</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">df</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.RandomState.standard_t" title="Link to this definition">#</a></dt>
<dd><p>Draw samples from a standard Student’s t distribution with <em class="xref py py-obj">df</em> degrees
of freedom.</p>
<p>A special case of the hyperbolic distribution. As <em class="xref py py-obj">df</em> gets
large, the result resembles that of the standard normal
distribution (<a class="reference internal" href="numpy.random.RandomState.standard_normal.html#numpy.random.RandomState.standard_normal" title="numpy.random.RandomState.standard_normal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">standard_normal</span></code></a>).</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>New code should use the <a class="reference internal" href="numpy.random.Generator.standard_t.html#numpy.random.Generator.standard_t" title="numpy.random.Generator.standard_t"><code class="xref py py-obj docutils literal notranslate"><span class="pre">standard_t</span></code></a>
method of a <a class="reference internal" href="../generator.html#numpy.random.Generator" title="numpy.random.Generator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Generator</span></code></a> instance instead;
please see the <a class="reference internal" href="../index.html#random-quick-start"><span class="std std-ref">Quick start</span></a>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>df</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Degrees of freedom, must be > 0.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape. If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n,</span> <span class="pre">k)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn. If size is <code class="docutils literal notranslate"><span class="pre">None</span></code> (default),
a single value is returned if <code class="docutils literal notranslate"><span class="pre">df</span></code> is a scalar. Otherwise,
<code class="docutils literal notranslate"><span class="pre">np.array(df).size</span></code> samples are drawn.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>Drawn samples from the parameterized standard Student’s t distribution.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.random.Generator.standard_t.html#numpy.random.Generator.standard_t" title="numpy.random.Generator.standard_t"><code class="xref py py-obj docutils literal notranslate"><span class="pre">random.Generator.standard_t</span></code></a></dt><dd><p>which should be used for new code.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The probability density function for the t distribution is</p>
<div class="math notranslate nohighlight">
\[P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
\Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}\]</div>
<p>The t test is based on an assumption that the data come from a
Normal distribution. The t test provides a way to test whether
the sample mean (that is the mean calculated from the data) is
a good estimate of the true mean.</p>
<p>The derivation of the t-distribution was first published in
1908 by William Gosset while working for the Guinness Brewery
in Dublin. Due to proprietary issues, he had to publish under
a pseudonym, and so he used the name Student.</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="r89f5270d198b-1" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">1</a><span class="fn-bracket">]</span></span>
<p>Dalgaard, Peter, “Introductory Statistics With R”,
Springer, 2002.</p>
</div>
<div class="citation" id="r89f5270d198b-2" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<p>Wikipedia, “Student’s t-distribution”
<a class="reference external" href="https://en.wikipedia.org/wiki/Student's_t-distribution">https://en.wikipedia.org/wiki/Student’s_t-distribution</a></p>
</div>
</div>
<p class="rubric">Examples</p>
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<p>From Dalgaard page 83 <a class="reference internal" href="#r89f5270d198b-1" id="id3">[1]</a>, suppose the daily energy intake for 11
women in kilojoules (kJ) is:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">intake</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">5260.</span><span class="p">,</span> <span class="mi">5470</span><span class="p">,</span> <span class="mi">5640</span><span class="p">,</span> <span class="mi">6180</span><span class="p">,</span> <span class="mi">6390</span><span class="p">,</span> <span class="mi">6515</span><span class="p">,</span> <span class="mi">6805</span><span class="p">,</span> <span class="mi">7515</span><span class="p">,</span> \
<span class="gp">... </span> <span class="mi">7515</span><span class="p">,</span> <span class="mi">8230</span><span class="p">,</span> <span class="mi">8770</span><span class="p">])</span>
</pre></div>
</div>
<p>Does their energy intake deviate systematically from the recommended
value of 7725 kJ? Our null hypothesis will be the absence of deviation,
and the alternate hypothesis will be the presence of an effect that could be
either positive or negative, hence making our test 2-tailed.</p>
<p>Because we are estimating the mean and we have N=11 values in our sample,
we have N-1=10 degrees of freedom. We set our significance level to 95% and
compute the t statistic using the empirical mean and empirical standard
deviation of our intake. We use a ddof of 1 to base the computation of our
empirical standard deviation on an unbiased estimate of the variance (note:
the final estimate is not unbiased due to the concave nature of the square
root).</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">intake</span><span class="p">)</span>
<span class="go">6753.636363636364</span>
<span class="gp">>>> </span><span class="n">intake</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">ddof</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1142.1232221373727</span>
<span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">intake</span><span class="p">)</span><span class="o">-</span><span class="mi">7725</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">intake</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">ddof</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">intake</span><span class="p">)))</span>
<span class="gp">>>> </span><span class="n">t</span>
<span class="go">-2.8207540608310198</span>
</pre></div>
</div>
<p>We draw 1000000 samples from Student’s t distribution with the adequate
degrees of freedom.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<span class="gp">>>> </span><span class="n">s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">standard_t</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">1000000</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">h</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>
<p>Does our t statistic land in one of the two critical regions found at
both tails of the distribution?</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o"><</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s</span><span class="p">))</span> <span class="o">/</span> <span class="nb">float</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
<span class="go">0.018318 #random < 0.05, statistic is in critical region</span>
</pre></div>
</div>
<p>The probability value for this 2-tailed test is about 1.83%, which is
lower than the 5% pre-determined significance threshold.</p>
<p>Therefore, the probability of observing values as extreme as our intake
conditionally on the null hypothesis being true is too low, and we reject
the null hypothesis of no deviation.</p>
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