document.write( "Question 430162: The density function of X is
\n" ); document.write( "f(x)={a+bx^2 if 0<=x<=1
\n" ); document.write( " 0 otherwise\r
\n" ); document.write( "\n" ); document.write( "the expectation of x is E(x)=3.75
\n" ); document.write( "find a and b \n" ); document.write( "

Algebra.Com's Answer #298898 by robertb(5830) $\"\"$ $\"About$

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The area under the density curve must be 1:\r
\n" ); document.write( "\n" ); document.write( " $\"int%28%28a%2Bbx%5E2%29%2C+dx%2C+0%2C1%29+=+1\"$ \r
\n" ); document.write( "\n" ); document.write( "The expectation is 3.75:\r
\n" ); document.write( "\n" ); document.write( " $\"int%28x%28a%2Bbx%5E2%29%2C+dx%2C+0%2C1%29+=+3.75\"$ , or $\"int%28%28ax%2Bbx%5E3%29%2C+dx%2C+0%2C1%29+=+3.75\"$ .\r
\n" ); document.write( "\n" ); document.write( "Upon integration of the first, we get $\"a%2Bb%2F3+=+1\"$ , or $\"a+=+1-b%2F3\"$ .
\n" ); document.write( "Upon integration of the second, we get $\"a%2F2+%2B+b%2F4+=+3.75\"$ , or\r
\n" ); document.write( "\n" ); document.write( " $\"1%2F2+-+b%2F6+%2B+b%2F4+=+3.75\"$ after substitution.
\n" ); document.write( "==> b = 39
\n" ); document.write( "==> a = 1-39/3 = -12. \n" ); document.write( "

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