document.write( "Question 334471: A random variable X can have values -4,-1,2,3 and 4, each with probability 0.2 .
\n" ); document.write( " find a) the density function
\n" ); document.write( " b) the mean and
\n" ); document.write( " c) the variance of the random Y = 3x^3 \n" ); document.write( "

Algebra.Com's Answer #239721 by jrfrunner(365) $\"\"$ $\"About$

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a) density function:
\n" ); document.write( "X=-4 p(X)=0.2
\n" ); document.write( "X=-1 p(X)=0.2
\n" ); document.write( "X=2 p(X)=0.2
\n" ); document.write( "X=3 p(X)=0.2
\n" ); document.write( "X=4 p(X)=0.2
\n" ); document.write( "all other X, P(X)=0
\n" ); document.write( "---
\n" ); document.write( "b) mean= $\"mu\"$
\n" ); document.write( " = E(X)= $\"sum+%28x%2Ap%28x%29%29\"$ =-4*(.2)-1*(0.2)+2*(0.2)+3*(0.2)+4*(0.2)=0.8
\n" ); document.write( "---
\n" ); document.write( "c) the variance of the random Y = 3x^3
\n" ); document.write( " $\"V%28Y%29=sigma%5E2=E%28Y%5E2%29-E%5E2%28Y%29=E%28y%5E2%29-mu%5E2\"$
\n" ); document.write( "= $\"V%283%2Ax%5E3%29=9%2AV%28X%5E3%29=9%2A%28E%28%28x%5E3%29%5E2%29-E%5E2%28x%5E3%29%29\"$
\n" ); document.write( "= $\"9%2A%28E%28x%5E6%29-E%5E2%28x%5E3%29%29\"$
\n" ); document.write( "--
\n" ); document.write( " $\"E%28X%5E6%29=sum%28x%5E6%2Ap%28x%29%29\"$ =(-4)^6*(0.2)+(-1)^6*(0.2)+2^6*(0.2)+3^6*(0.2)+4^6*(0.2)=1797.2
\n" ); document.write( "--
\n" ); document.write( " $\"mu=E%28X%5E3%29=sum%28x%5E3%2Ap%28x%29%29\"$ =(-4)^3*(0.2)+(-1)^3*(0.2)+2^3*(0.2)+3^3*(0.2)+4^3*(0.2)=6.8
\n" ); document.write( "--
\n" ); document.write( " $\"mu%5E2=E%5E2%28X%5E3%29=%286.8%29%5E2=46.24\"$
\n" ); document.write( "--
\n" ); document.write( " $\"V%28Y%29=sigma%5E2=9%2A%28E%28x%5E6%29-E%5E2%28x%5E3%29%29=9%2A%281797.2-46.24%29=15758.64\"$ \n" ); document.write( "

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