document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252226>Question 252226</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let the random variable X have the density function\r<BR>\n" );
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document.write( " f(x)= {kx                    for 0<x<sqrt(2/k) <BR>\n" );
document.write( "       {0                     elsewhere \r<BR>\n" );
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document.write( "If the mode of this distribution is at x =sqrt(2)/4, then what is the median of X?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #185130 by </B><A HREF=/tutors/aboutme.mpl?userid=Greenfinch><B>Greenfinch(383)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Greenfinch><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Greenfinch><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=185130\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For a continuous random variable, the probability density function is<BR>\n" );
document.write( "f(x) = { kx for 0 < x < sqrt (2/k)<BR>\n" );
document.write( "       {  else   0<BR>\n" );
document.write( "The cumulative distribution function F(x) for this is therefore the integral from 0 to sqrt (2/k) of kx dx or [(k x^2)/2] from  x = 0 to sqrt(2/k).<BR>\n" );
document.write( "The graph of the function starts at 0 when x = 0 and has a slope of k. The highest point is therefore at the point sqrt(2/k) which is the mode. The shape is a triangle whose length is sqrt(2/k) and whose height is k x sqrt(2/k). This area is the total probability and is therefore equal to 1. Since the k appears in both parts, this cannot be used to determine k, but knowing that the mode is at sqrt (2)/4 gives sqrt( k) = 4 so k = 16. The pdf now becomes<BR>\n" );
document.write( "f(x) = 16x for {0 < x < sqrt (1/8)<BR>\n" );
document.write( "               { else <BR>\n" );
document.write( "The median occurs when the cdf F(x) = 1/2<BR>\n" );
document.write( "so F(x)= 8 x^2 = 1/2<BR>\n" );
document.write( "x^2 = 1/16<BR>\n" );
document.write( "x= 1/4<BR>\n" );
document.write( "This fits in with expectations, The median is 1/sqrt2 = .7071 of the distance along the x axis. </SPAN>\n" );
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