6496
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1
2011-02-01T00:00:01
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<li><a href="http://nrich.maths.org/6149">Warm-up</a></li>
<li><a href="http://nrich.maths.org/7574">Try this next</a></li>
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<li><a href="http://understandinguncertainty.org/node/84">Read: mathematics</a></li>
<li><a href="http://plus.maths.org/content/os/latestnews/may-aug10/gene_package/index">Read: science</a></li>
<li><a href="http://plus.maths.org/content/teacher-package-statistics-and-probability-theory">Explore further</a></li>
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<p>I have a series of six probability density functions $X$, each of which satisfies at least one of the following conditions:<br></br>
</p>
<ol>
<li>$X$ is non-negative.</li>
<li>$X$ can take any positive value.</li>
<li>$X$ is a reasonably realistic pdf for a model of a share price of a bank in a year's time.</li>
<li>The probability of $X$ taking a value in the range $(a, b)$ is the same as the chance of $X$ taking a value in the range $(-b, -a)$.</li>
<li>$X$ is a reasonably realistic pdf for a model of human life expectancy.</li>
<li>There is a number $a$ such that $P(x> a) \geq 2 P(x> 2a)$.</li>
</ol>
<br></br>
<p>Which of the following six curves (ignoring scale) would be the potential candidates for these mathematical descriptions? Are multiple matches possible? What axes and scales would you choose in each case?<br></br>
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<mdo:image alt="" height="330" src="pdfs.jpg" width="442"></mdo:image><br></br>
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Part 1:<br></br>
<br></br>
Statement 1: All<br></br>
<br></br>
Statement 2:None<br></br>
<br></br>
Statement 3: Curves 2 and 3<br></br>
<br></br>
Statement 4: Curves 4 and 5<br></br>
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Statement 5: Curve 2<br></br>
<br></br>
Statement 6: Curve 2, 3, 4 and 6<br></br>
<br></br>
<br></br>
Curve A:<br></br>
IV characteristics of an ohmic device<br></br>
<br></br>
Curve B:<br></br>
<br></br>
Curve C:<br></br>
<br></br>
Curve D:<br></br>
<br></br>
Curve E:<br></br>
This curve appears to be the integral of a sinusoid. It could
therefore represent the total electrical power dissipated as a
function of time for a device through which we pass an alternating
current<br></br>
<br></br>
Curve F:<br></br>
This curve could represent the amplitude of oscillation of an
undammed spring which is the suddenly stopped.<br></br>
<br></br>
Curve G:<br></br>
This curve could represent the magnitude of a pendulums
displacement from its equilibrium position (assuming the pendulum
begins at maximum displacement).<br></br>
<br></br>
Curve H:<br></br>
Charging of a capacitor<br></br>
<br></br>
Curve I:<br></br>
Radioactive decay. This curve could represent the number of nuclei
decaying per second from some radioactive source.<br></br>
<br></br>
Curve J:<br></br>
Normal distribution: Height or weight<br></br>
This could also be a resonance curve. The Y axis could be the
amplitude of osciallation and the X axis could represent the
frequency of the input, at the natural frequency we would get a
very high amplitude response.<br></br>
<br></br>
Curve K:<br></br>
<br></br>
<br></br>
Curve L:<br></br>
This curve could represent the fracture toughness of an alloyed
metal, the Y axis representing the fracture toughness and the x -
axis representing the volume fraction of the alloy.<br></br></mdoxml>
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