Example: Seed for Random Number Generation

Functions > Statistics > Probability Distributions > Example: Seed for Random Number Generation

Use the Seed function to control the process of generating new sets of random numbers.

1. Use functions Seed and rnorm in a short program to generate identical sets of normally distributed random numbers for three iterations by resetting the Seed value.

<region id="ID0EQHVBB" top="268.80000000000007" left="38.400000000000006" height="117.15199999999999" width="372.98" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="84">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">SameNormal</id>
        <boundVars>
          <id labels="VARIABLE" xml:space="preserve">p</id>
          <id labels="VARIABLE" xml:space="preserve">μ</id>
          <id labels="VARIABLE" xml:space="preserve">σ</id>
        </boundVars>
      </function>
      <program>
        <for>
          <id labels="VARIABLE" xml:space="preserve">i</id>
          <range>
            <real>0</real>
            <real>2</real>
          </range>
          <program>
            <apply>
              <id labels="FUNCTION" xml:space="preserve">Seed</id>
              <real>1</real>
            </apply>
            <localDefine>
              <id labels="VARIABLE" xml:space="preserve">nums</id>
              <apply>
                <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">rnorm</id>
                <sequence>
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">p</id>
                  <id labels="VARIABLE" xml:space="preserve">μ</id>
                  <id labels="VARIABLE" xml:space="preserve">σ</id>
                </sequence>
              </apply>
            </localDefine>
            <localDefine>
              <apply>
                <matcol />
                <id labels="VARIABLE" xml:space="preserve">M</id>
                <id labels="VARIABLE" xml:space="preserve">i</id>
              </apply>
              <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">nums</id>
            </localDefine>
          </program>
        </for>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M</id>
      </program>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• The same set of random numbers is generated each time because the seed is set to 1.

• SameNormal returns a matrix of three identical columns, one for each iteration.

2. Use functions hist and max to plot a histogram of the second random set with n_bins bins. Add a horizontal marker to show the maximum of the returned set.

<region id="ID0E5OVBB" top="691.2" left="38.400000000000006" height="45.2" width="228.19666666666669" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="94">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <indexer />
        <id labels="VARIABLE" xml:space="preserve">S1</id>
        <id labels="*" xml:space="preserve">q</id>
      </apply>
      <apply>
        <indexer />
        <parens>
          <apply>
            <matcol />
            <apply>
              <id labels="VARIABLE" xml:space="preserve">SameNormal</id>
              <sequence>
                <id labels="VARIABLE" xml:space="preserve">p</id>
                <id labels="VARIABLE" xml:space="preserve">μ</id>
                <id labels="VARIABLE" xml:space="preserve">σ</id>
              </sequence>
            </apply>
            <real>1</real>
          </apply>
        </parens>
        <id xml:space="preserve">q</id>
      </apply>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

<region id="ID0EIQVBB" top="921.60000000000014" left="38.400000000000006" height="235" width="575.781818181818" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white" origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="100">
          <traceStyle color="#FF9ABC66" symbol="none" line-weight="1" line-style="Solid">columns</traceStyle>
        </trace>
      </traces>
      <graph-size width="456.581818181818" height="177.4" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="20">
          <axisLine position="origin" positionticmark="0" legendWidth="92.1466666666667" />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="107">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">range</id>
              </math>
              <math resultRef="108">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="102">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="104">
              <real xmlns="http://schemas.mathsoft.com/math50">5</real>
            </secondTickValue>
            <endValue resultRef="106">
              <real xmlns="http://schemas.mathsoft.com/math50">20</real>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="150">
          <axisLine position="origin" positionticmark="0" legendWidth="87.9033333333333" />
          <axisGrid>
            <gridFrequency>7</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers>
            <marker resultRef="109" display="true" color="#FFED1D2F" line-style="Dash" value="129">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <id xml:space="preserve">max</id>
                <id xml:space="preserve">same</id>
              </apply>
            </marker>
          </markers>
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="117">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">same</id>
              </math>
              <math resultRef="118">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="112">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="114">
              <real xmlns="http://schemas.mathsoft.com/math50">25</real>
            </secondTickValue>
            <endValue resultRef="116">
              <real xmlns="http://schemas.mathsoft.com/math50">150</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

3. Modify the program to generate three different sets of normally distributed random numbers by incrementing the seed value with each iteration.

<region id="ID0EYRVBB" top="1267.2000000000003" left="38.400000000000006" height="117.15199999999999" width="365.72" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="130">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">DiffNormal</id>
        <boundVars>
          <id labels="VARIABLE" xml:space="preserve">p</id>
          <id labels="VARIABLE" xml:space="preserve">μ</id>
          <id labels="VARIABLE" xml:space="preserve">σ</id>
        </boundVars>
      </function>
      <program>
        <for>
          <id labels="VARIABLE" xml:space="preserve">i</id>
          <range>
            <real>0</real>
            <real>2</real>
          </range>
          <program>
            <apply>
              <id labels="FUNCTION" xml:space="preserve">Seed</id>
              <apply>
                <plus />
                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">i</id>
                <real>1</real>
              </apply>
            </apply>
            <localDefine>
              <id labels="VARIABLE" xml:space="preserve">nums</id>
              <apply>
                <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">rnorm</id>
                <sequence>
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">p</id>
                  <id labels="VARIABLE" xml:space="preserve">μ</id>
                  <id labels="VARIABLE" xml:space="preserve">σ</id>
                </sequence>
              </apply>
            </localDefine>
            <localDefine>
              <apply>
                <matcol />
                <id labels="VARIABLE" xml:space="preserve">M</id>
                <id labels="VARIABLE" xml:space="preserve">i</id>
              </apply>
              <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">nums</id>
            </localDefine>
          </program>
        </for>
        <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M</id>
      </program>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• Now that the seed value changes with each iteration, each set of random numbers is unique.

• Since iteration=0 of both programs starts with seed=1, column=0 of both returned matrices contain identical sets.

4. Extract the second data set from the returned matrix:

<region id="ID0EGWVBB" top="1593.6000000000001" left="38.400000000000006" height="45.2" width="220.93666666666667" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="132">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <indexer />
        <id labels="VARIABLE" xml:space="preserve">S2</id>
        <id labels="*" xml:space="preserve">q</id>
      </apply>
      <apply>
        <indexer />
        <parens>
          <apply>
            <matcol />
            <apply>
              <id labels="VARIABLE" xml:space="preserve">DiffNormal</id>
              <sequence>
                <id labels="VARIABLE" xml:space="preserve">p</id>
                <id labels="VARIABLE" xml:space="preserve">μ</id>
                <id labels="VARIABLE" xml:space="preserve">σ</id>
              </sequence>
            </apply>
            <real>1</real>
          </apply>
        </parens>
        <id xml:space="preserve">q</id>
      </apply>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

5. Use function hist to generate a histogram of the random set with n_bins bins, then calculate its maximum.

6. Plot the histogram and use a horizontal marker to show its maximum:

<region id="ID0EYZVBB" top="1785.6000000000004" left="38.400000000000006" height="235" width="575.781818181818" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white" origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="138">
          <traceStyle color="#FFF47D44" symbol="none" line-weight="1" line-style="Dash">columns</traceStyle>
        </trace>
      </traces>
      <graph-size width="456.581818181818" height="177.4" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="20">
          <axisLine position="origin" positionticmark="0" legendWidth="92.1466666666667" />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="145">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">range</id>
              </math>
              <math resultRef="146">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="140">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="142">
              <real xmlns="http://schemas.mathsoft.com/math50">5</real>
            </secondTickValue>
            <endValue resultRef="144">
              <real xmlns="http://schemas.mathsoft.com/math50">20</real>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="175">
          <axisLine position="origin" positionticmark="0" legendWidth="80.32" />
          <axisGrid>
            <gridFrequency>8</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers>
            <marker resultRef="147" display="true" color="#FF662D91" line-style="Dash" value="155">
              <apply xmlns="http://schemas.mathsoft.com/math50">
                <id xml:space="preserve">max</id>
                <id xml:space="preserve">diff</id>
              </apply>
            </marker>
          </markers>
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="155">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">diff</id>
              </math>
              <math resultRef="156">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="150">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <secondTickValue resultRef="152">
              <real xmlns="http://schemas.mathsoft.com/math50">25</real>
            </secondTickValue>
            <endValue resultRef="154">
              <real xmlns="http://schemas.mathsoft.com/math50">175</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>