document.write( "Question 1203810: A random variable X has the pdf\r
\n" ); document.write( "\n" ); document.write( " x^2 if 0 < x ≤ 1
\n" ); document.write( "f(x) = 2/3 if 1 < x ≤ 2
\n" ); document.write( " 0 otherwise\r
\n" ); document.write( "\n" ); document.write( "(a)Find the median of X.
\n" ); document.write( "(b)Sketch the graph of the CDF and show the position of the median on the graph. \n" ); document.write( "

Algebra.Com's Answer #839942 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "That's the graph.  Now we'll shade the area under it:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we find the area between the graph and the x-axis, take half\r\n" );
document.write( "of it and find the value of x which divides the entire area into\r\n" );
document.write( "two parts.\r\n" );
document.write( "\r\n" );
document.write( "First we find the area under the curved (parabola) part\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The right part of the graph is just a rectangle with base 1 and height , \r\n" );
document.write( "so its area is just .\r\n" );
document.write( "\r\n" );
document.write( "The entire shaded area is \r\n" );
document.write( "\r\n" );
document.write( "So we need to find the value of x such that half the area  is left of\r\n" );
document.write( "it and the other half , is on the right of it.\r\n" );
document.write( "\r\n" );
document.write( "The area under the parabola is  so that leaves  the we must take \r\n" );
document.write( "of the rectangle. To get the base of a rectangle with area  that has height , \r\n" );
document.write( "we divide the area by the height and get .  So the\r\n" );
document.write( "median is  on a unit past 1, or .\r\n" );
document.write( "\r\n" );
document.write( "So the median is . To indicate the median, we can draw a green\r\n" );
document.write( "vertical line through the graph at : \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "

\n" );