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Var[a(X^2)+b] = E[ ( a(X^2) + b - (aE[X^2] + b ) )^2 ]

= E[ ( a(X^2) + b -aE[X^2] - b )^2 ]

= E[ ( a(X^2) - aE[X^2] )^2]

= E[ (a^2) ( (X^2) -E[X^2] )^2 ]

= E[ ( (X^2) - E[X^2] ) ( (X^2) - E[X^2] ) ]

= (a^2) E[ (X^4) -2(X^2)E[X^2] + ( E[X^2] )^2]

= (a^2) { E[X^4] -2E[X^2]E[X^2] + ( E[X^2] )^2 }

= (a^2) { E[X^4] -2( E[X^2] )^2 + ( E[X^2] )^2 }

= (a^2) { E[X^4] - ( E[X^2] )^2 }

= (a^2) Var[X^2]

*however, we still do not know what Var[X^2] is....

Q: What is the variance of X squared?

This answer is:

Related answers

Var[a(X^2)+b] = E[ ( a(X^2) + b - (aE[X^2] + b ) )^2 ]

= E[ ( a(X^2) + b -aE[X^2] - b )^2 ]

= E[ ( a(X^2) - aE[X^2] )^2]

= E[ (a^2) ( (X^2) -E[X^2] )^2 ]

= E[ ( (X^2) - E[X^2] ) ( (X^2) - E[X^2] ) ]

= (a^2) E[ (X^4) -2(X^2)E[X^2] + ( E[X^2] )^2]

= (a^2) { E[X^4] -2E[X^2]E[X^2] + ( E[X^2] )^2 }

= (a^2) { E[X^4] -2( E[X^2] )^2 + ( E[X^2] )^2 }

= (a^2) { E[X^4] - ( E[X^2] )^2 }

= (a^2) Var[X^2]

*however, we still do not know what Var[X^2] is....

View page

var(x) = E[(x - E(x))2]

= E[(x - E(x)) (x - E(x))] <-------------Expand into brackets

= E[x2 - xE(x) - xE(x) + (E(x))2] <---Simplify

= E[x2 - 2xE(x) + (E(x))2]

= E(x2) + E[-2xE(x)] + E[(E(x))2]

= E(x2) - 2E[xE(x)] + E[(E(x))2] <---Bring (-2) constant outside

= E(x2) - 2E(x)E[E(x)] + E[(E(x))2] <--- E[xE(x)] = E(x)E(x)

= E(x2) - 2E(x)E(x) + [E(x)]2 <----------E[E(x)] = E(x)

= E(x2) - 2[E(x)]2 + [E(x)]2

var(x) = E(x2) - [E(x)]2

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The power law of indices says:

(x^a)^b = x^(ab) = x^(ba) = (x^b)^a

→ e^(2x) = (e^x)²

but e^x = 2

→ e^(2x) = (e^x)² = 2² = 4

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