PDFofDerivedRV - Y = X 2 where X is a continuous random...

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How to Evaluate the Probability Functions for a Derived Continuous Random Variable Let X be a continuous random variable. Then Y = g ( X ), where g ( · ) is a continuous function, will be called a derived random variable from X or a function of random variable of X . For example let X be a continuous random variable with probability density function (PDF) f X ( x ) = 1 0 < x < 1 0 elsewhere . Here we used f X ( x ) instead of f ( x ) to distinguish the PDF of X from that of Y . Then Y 1 = 5 X and Y 2 = X 3 would be derived random variables from X . In order to find the PDF of Y , we follow a two-step procedure: Step 1 Find the cumulative distribution function (CDF) of Y , i.e., F Y ( y ) = P [ Y y ]. Again, we used a subscript Y here to distinguish the distribution for Y from that for X . Step 2 Take the derivative of F Y ( y ) with respect of y to get the PDF for y . Specifically we utilize the following relationship. f Y ( y ) = dF Y ( y ) dy . Let us take a look at one example. Assume that we are interested in finding out the PDF for
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Unformatted text preview: Y = X 2 where X is a continuous random variable with PDF shown above. Then Y is a continuous random variable with a range of (0,1). As we discussed earlier in the classroom, the CDF of Y carries the same amount of information as the PDF of Y . Let us first evaluate the probability P [ Y ≤ . 5]. F Y (0 . 5) = P [ Y ≤ . 5] = P [ X 2 ≤ . 5] = P [ X ≤ √ . 5] = Z √ . 5-∞ f ( x ) dx = Z √ . 5 1 dx = √ . 5 . We can replace 0 . 5 by a general y to get F Y ( y ) = √ y where 0 < y < 1. In summary, we have F Y ( y ) =          y ≤ √ y < y < 1 1 y ≥ 1 . Taking the derivative of F Y ( y ) with respect to y , we obtain the PDF for Y , i.e., f Y ( y ) =    1 2 y-1 / 2 < y < 1 elsewhere ....
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This note was uploaded on 01/19/2012 for the course IEN 311 taught by Professor Qiu during the Spring '11 term at University of Miami.

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