6420
/www/nrich/html/content/id/6420/
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2011-02-01T00:00:01
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<p>A random variable $X$ has a zero probability of taking non-positive values but has a non-zero probability of taking values in any range $[0, x]$ for any $x> 0$. The curve describing the probability density function forms an arc of a circle. Which of these are possible shapes (ignoring the scale) for the probability density function $f(x)$? Identify clearly the mathematical reasons, using
the correct terminology, for your answers.<br></br>
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<mdo:image alt="" height="191" src="image.jpg" width="367"></mdo:image><br></br>
If the radius of the circle forming the arc of the pdf is $1$, what is the maximum value that the random variable could possibly take?<br></br>
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Which of the other arcs are possible candidates for probability density functions? Can you invent mathematical scenarios which would lead to these pdfs?<br></br>
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<h3>Why do this problem?</h3>
<p>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=6420&part=">problem</a> gives an opportunity to explore the properties of pdfs using the mathematics of sectors of circles.</p>
<h3>Possible approach</h3>
<div>There are two main parts to this problem.</div>
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<div>The first is to understand why certain shapes are firstly valid pdfs and secondly how they satisfy the technical requirement of the question. This would benefit from a discussion approach.</div>
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<div>The second part, calculating the maximum value, will lead students into the mathematics of sectors and segments of circles.</div>
<h3>Key questions</h3>
<div>What properties must a pdf have?</div>
<div>How would the requirements of the questions relate to a graph?</div>
<div>In order to obtain the maximum possible value for the case of the circle of radius $1$, what do we know about the arc?</div>
<h3>Possible extension</h3>
<p>Try <a href="http://nrich.maths.org/public/viewer.php?obj_id=6141&part=">Into the Exponential Distribution</a>.</p>
<h3>Possible support</h3>
<div>Point students in the direction of the formula for the segment of a circle and that the area must be 1.</div>
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<mdoxml version="1.0">Think what properties probability density functions must
have.<br></br>
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You will need to use the formula for the area of a sector of a
circle.<br></br></mdoxml>
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Circle \pdf
What happens if this pdf is the arc of a circle?
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Probability density functions