Question

I. If a variable X has the mean μ and the variance σ, express the following quantities by (1.2) E3X+1] (1.4) E2x+5x+7] (1.5) V[3X] (1.6 VI2x-3]

Answer 1

Solution : ( 1.1 )

$\mathrm{E[2X]}=2*\mathrm{E[X]}=2\mu$

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Solution : ( 1.2 )

$\mathrm{E[3X+1]}=3*\mathrm{E[X]}+1=3\mu +1$

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Solution : ( 1.3 )

$\mathrm{E[X^{2}]=V[X]+(E[X])^{2}}=\sigma ^{2}+(\mu )^{2}=\sigma ^{2}+\mu ^{2}$

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Solution : ( 1.4 )

$\mathrm{E[2X^{2}+5X+7]}=2*\mathrm{E[X^{2}]}+5*\mathrm{E[X]}+7$

$=2*(\mathrm{V[X]+(E[X])^{2}})+5*\mathrm{E[X]}+7$

$=2*(\mathrm{\sigma ^{2}+(\mu )^{2}})+5*\mu+7$

$=2\mathrm{\sigma ^{2}+2\mu^{2}}+5\mu+7$

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Solution : ( 1.5 )

$\mathrm{V[3X]}=3^{2}*\mathrm{V[X]}=9*\mathrm{V[X]}=9\sigma ^{2}$

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Solution : ( 1.6 )

$\mathrm{V[2X-3]}=2^{2}*\mathrm{V[X]}=4*\mathrm{V[X]}=4\sigma ^{2}$