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Question Number 87643 by Ar Brandon last updated on 05/Apr/20
![Given a random variable X of image set X(Ω)=[1;−1;2] with probabilities P(X=1)=e^a , P(X=−1)=e^b , and P(X=2)=e^c where a, b, and c are in an Arithmetic Progression. Assuming the mathematical expection E(X) of X is equal to 1. 1∙ Suppose the common difference of the Arithmetic Progression is r a∙ Determine the exact values of the real numbers a, b, and c. b. Show that the variance V(X) of X is equal to ((22)/7).](Q87643.png)
$${Given}\:{a}\:{random}\:{variable}\:\boldsymbol{{X}}\:{of}\:{image}\:{set} \\ $$$${X}\left(\Omega\right)=\left[\mathrm{1};−\mathrm{1};\mathrm{2}\right]\:{with}\:{probabilities}\:{P}\left({X}=\mathrm{1}\right)={e}^{{a}} , \\ $$$${P}\left({X}=−\mathrm{1}\right)={e}^{{b}} ,\:{and}\:{P}\left({X}=\mathrm{2}\right)={e}^{{c}} \:{where}\:{a},\:{b},\:{and}\:{c}\: \\ $$$${are}\:{in}\:\:{an}\:{Arithmetic}\:{Progression}. \\ $$$${Assuming}\:{the}\:{mathematical}\:{expection}\:{E}\left({X}\right)\:{of}\:{X}\: \\ $$$${is}\:{equal}\:{to}\:\mathrm{1}. \\ $$$$\mathrm{1}\centerdot\:{Suppose}\:{the}\:{common}\:{difference}\:{of}\:{the}\:{Arithmetic}\:{Progression}\:{is}\:\boldsymbol{{r}} \\ $$$$\boldsymbol{{a}}\centerdot\:{Determine}\:{the}\:{exact}\:{values}\:{of}\:{the}\:{real}\:{numbers}\:{a},\:{b},\:{and}\:{c}. \\ $$$$\boldsymbol{{b}}.\:{Show}\:{that}\:{the}\:{variance}\:{V}\left({X}\right)\:{of}\:{X}\:{is}\:{equal}\:{to}\:\frac{\mathrm{22}}{\mathrm{7}}. \\ $$$$ \\ $$$$ \\ $$