Question 334471
a) density function:
X=-4 p(X)=0.2
X=-1 p(X)=0.2
X=2  p(X)=0.2
X=3  p(X)=0.2
X=4  p(X)=0.2
all other X, P(X)=0
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b) mean={{{mu}}}
 = E(X)={{{sum (x*p(x))}}}=-4*(.2)-1*(0.2)+2*(0.2)+3*(0.2)+4*(0.2)=0.8
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c) the variance of the random Y = 3x^3
{{{V(Y)=sigma^2=E(Y^2)-E^2(Y)=E(y^2)-mu^2}}}
={{{V(3*x^3)=9*V(X^3)=9*(E((x^3)^2)-E^2(x^3))}}}
={{{9*(E(x^6)-E^2(x^3))}}}    
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{{{E(X^6)=sum(x^6*p(x))}}}=(-4)^6*(0.2)+(-1)^6*(0.2)+2^6*(0.2)+3^6*(0.2)+4^6*(0.2)=1797.2
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{{{mu=E(X^3)=sum(x^3*p(x))}}}=(-4)^3*(0.2)+(-1)^3*(0.2)+2^3*(0.2)+3^3*(0.2)+4^3*(0.2)=6.8
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{{{mu^2=E^2(X^3)=(6.8)^2=46.24}}}
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{{{V(Y)=sigma^2=9*(E(x^6)-E^2(x^3))=9*(1797.2-46.24)=15758.64}}}