11. For the following discrete probability distribution: egin{tabular}{|c|l|l|l|l|l|l|} hline( x ) & 0 & 1 & 2 & 3 & 4 & 5 \ hline( P(x) ) & 0.2 & 0.05 & 0.1 & 0.3 & 0.2 & 0.15 \ hline end{tabular} Find the following: a. ( P(x<3) ) b. ( P(x ) between 1 and 4( ) ) c. ( P(x geq 3) ) d. ( P(x>2) ) e. ( E V(x) ) f. Variance and standard deviation 12. What production level will maximize profits for a company with cost and demand functions [ C(x)=3800+5 x-frac{x^{2}}{100} ext { and } R(x)=50-frac{x}{100} ] 13. Solve for ( x, y ) and ( z+ ) [ egin{array}{c} 2 x-y+z=2 \ 3 x+2 y+2 z=9 \ x+3 y-3 z=4 end{array} ] 14. A linear relationship is assumed in the following scenario: A machine produces 5 units and makes a profit of ( mathrm{P} 200 ). When the number of units is 15 , the profit is P500. Find the line equation and estimate the profit when 25 units are produced.
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The computer store has purchased three computers at 500 dollars apiece. It will sell them for 1000 dollars apiece. The manufacturer has agreed to repurchase any computers still unsold after a specified period at $200 apiece. Let X denote the number of computers sold, and suppose that x 0 1 2 3 P(X=x) 0.1 0.2 0.3 0.4 Let g(X) represent the profit associated with selling X units. a. Write the formula for the g(X) function and give the probability mass function table for the g(X). Then compute the expected value and the standard deviation of the g(X) function. b. If there are 10 computer stores like these, how much should we expect the aggregate of the profits of the 10 stores to be? If there are 3 computer stores like these, what is the probability that every single one of them makes less than 1000 dollars? c. What would be the value of E(X^3)? d. Write the moment generating function of X.
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