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197 lines
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197 lines
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HTML
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<title>Random Variables - Continuous</title>
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<meta name="description" content="A Random Variable is a set of possible values from a random experiment.">
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<div id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 style="text-align:center">Random Variables - Continuous</h1>
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<p>A Random Variable is a set of <b>possible values</b> from a random experiment.</p>
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<div class="example">
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<h3>Example: Tossing a coin: we could get Heads or Tails. </h3>
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<p> Let's give them the values <b>Heads=0</b> and <b>Tails=1</b> and we have a Random Variable "X":</p>
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<p> </p>
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<p class="center"><img src="images/random-variable-1.svg" alt="random variable 1"></p>
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<p>In short:</p>
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<p class="center larger">X = {0, 1} </p>
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<p>Note: We could choose Heads=100 and Tails=150 or other values if we want! It is our choice.</p>
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</div>
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<h2>Continuous</h2>
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<p>Random Variables can be either <a href="data-discrete-continuous.html"><span><span style="text-decoration:underline">Discrete
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or Continuous</span></span></a>:</p>
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<ul>
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<li>Discrete Data can only take certain values (such as 1,2,3,4,5) </li>
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<li>Continuous Data can take any value within a range (such as a person's height)</li>
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</ul>
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<p>In our Introduction to <span style="text-align:center"><a href="random-variables.html">Random Variables</a></span> (please read that first!) we look at many examples of Discrete Random Variables.</p>
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<p>But here we look at the more advanced topic of <span style="text-align:center">Continuous Random Variables. </span></p>
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<p> </p>
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<h2>The Uniform Distribution</h2>
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<p>The Uniform Distribution (also called the Rectangular Distribution) is the simplest distribution. </p>
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<p>It has equal probability for all values of the
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Random variable between <b>a</b> and <b>b</b>:</p>
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<p style="text-align:center"><img src="images/uniform-distribution.svg" alt="uniform distribution p=1/(b-a)"><br>
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The probability of any value between <b>a</b> and <b>b</b> is <b>p</b>
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</p>
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<p>We also know that p = 1/(b-a), because the total of all probabilities must
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be 1, so</p>
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<div class="so">the area of the rectangle = 1</div>
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<div class="so">p × (b−a) = 1</div>
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<div class="so">p = 1/(b−a)</div>
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<p>We can write:</p>
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<p class="center"><span class="larger">P(X = x) = 1/(b−a) for a ≤
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x ≤ b</span><br>
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P(X = x) = 0 otherwise </p>
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<div class="example">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/old-faithful.jpg" alt="old faithful" height="171" width="120"></p>
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<h3>Example: Old Faithful erupts every 91 minutes. You arrive there at random and wait for 20 minutes ... what is the probability you will see it erupt?</h3>
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<p> </p>
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<p>This is actually easy to calculate, 20 minutes out of 91 minutes is: </p>
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<p class="center"><b>p = 20/91 = 0.22</b> (to 2 decimals)</p>
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<p> </p>
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<p>But let's use the Uniform Distribution for practice.</p>
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<p>To find the probability between <b>a</b> and <b>a+20</b>, find the blue area:</p>
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<p class="center"><img src="images/uniform-distribution-ex.svg" alt="uniform distribution example"></p>
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<p class="center larger">Area = (1/91) x (a+20 − a) <br>
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= (1/91) x 20 <br>
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= 20/91 <br>
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= <b>0.22</b> (to 2 decimals) </p>
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<p>So there is a 0.22 probability you will see Old Faithful erupt.</p>
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<p> </p>
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<p>If you waited the full 91 minutes you would be sure (<b>p=1</b>) to have seen it erupt.</p>
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<p>But remember this is a random thing! It might erupt the moment you arrive, or any time in the 91 minutes.</p>
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</div>
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<h2>Cumulative Uniform Distribution</h2>
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<p>We can have the Uniform Distribution as a <b>cumulative</b> (adding up as it goes along) distribution: </p>
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<p class="center"><img src="images/uniform-distribution-cumulative.gif" alt="uniform distribution cumulative" height="171" width="190"><br>
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The probability starts at 0 and builds up to 1</p>
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<div class="words">
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<p>This type of thing is called a "Cumulative distribution function", often shortened to "CDF"</p>
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</div>
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<div class="example">
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<h3>Example (continued):</h3>
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<p>Let's use the "CDF" of the previous Uniform Distribution to work out the probability:</p>
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<p class="center"><img src="images/uniform-distribution-cumulative-ex.gif" alt="uniform distribution cumulative" height="169" width="203"><br>
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</p>
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<p>At <b>a+20</b> the probability has accumulated to about <b>0.22</b></p>
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</div>
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<h2>Other Distributions</h2>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td width="300">Knowing how to use the Uniform Distribution helps when dealing with more complicated distributions like this one:</td>
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<td><img src="images/non-uniform-distribution.gif" alt="non uniform distribution" height="170" width="190"></td>
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</tr>
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</tbody></table>
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<div class="words">
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<p>The general name for any of these is probability density function or "pdf"</p>
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</div>
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<h2>The Normal Distribution</h2>
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<p> The most important continuous distribution
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is the <a href="standard-normal-distribution.html">Standard Normal Distribution</a></p>
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<p>It is so important the Random Variable has its own special letter <b>Z</b>.</p>
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<p>The graph for Z is a symmetrical bell-shaped curve:</p>
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<p class="center"><img src="images/standard-normal-distribution-simple.gif" alt="standard normal distribution " height="231" width="423"></p>
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<p>Usually we want to find the probability
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of Z being between certain values.</p>
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<div class="example">
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<h3>Example: P(0 < Z < 0.45)</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/standard-normal-distribution-0-45.gif" alt="standard normal distribution 0.45 = 0.1736" height="163" width="178"></p>
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<p>(What is the probability that Z is between 0 and 0.45)</p>
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<p>This is found by using the <a href="standard-normal-distribution-table.html">Standard
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Normal Distribution Table</a></p>
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<p>Start at the row for 0.4, and read along
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until 0.45: there is the value 0.1736</p>
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<p class="center larger">P(0 < Z < 0.45) = 0.1736</p>
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</div>
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<p> </p>
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<h2>Summary</h2>
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<ul class="larger">
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<li>A <span style="font-weight:bold">Random
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Variable</span> is a variable whose possible values are numerical outcomes
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of a random experiment.</li>
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<li>Random
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Variables can be discrete or continuous.</li>
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<li>An
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important example of a continuous Random variable is the <span style="font-weight:bold">Standard Normal</span> variable, <b>Z</b>.</li>
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</ul>
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<p> </p>
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<div class="questions">
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<script>getQ(8868, 8869, 8870, 8871, 8872, 8873, 8874, 8875, 8876, 8877);</script>
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</div>
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<div class="related">
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<a href="random-variables.html">Random Variables</a>
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<a href="index.html">Data Index</a>
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