Answer:
The value is [tex]E(X) = \$ 1.7067 [/tex]
Step-by-step explanation:
From the question we are told that
The parameters are α = 3, θ = 0.5
The cost of making a unit on the first day is c = $2
The selling price of a unit on the first day is s = $5
The selling price of a leftover unit on the second day is v = $ 1
Generally the profit of a unit on the first day is
[tex]p_1 = 5 - 2[/tex]
[tex]p_1 = \$3 [/tex]
The profit of a unit on the second day is
[tex]p_2 = 1 - 2[/tex]
=> [tex]p_2 = - \$1 [/tex]
Generally the probability of making profit greater than $ 1 is mathematically represented as
[tex]P(X > 1 ) = Gamma (X ,\alpha , \theta)[/tex]
=> [tex]P(X > 1 ) = Gamma (1 ,3 , 0.5)[/tex]
Now from the gamma distribution table we have that
[tex]P(X > 1 ) = 0.67668[/tex]
Generally the probability of making profit less than or equal to $ 1 is mathematically represented as
[tex] P(X \le 1 ) = 1 - P(X > 1 )[/tex]
=> [tex] P(X \le 1 ) = 1 - 0.67668[/tex]
=> [tex] P(X \le 1 ) = 0.32332[/tex]
So the probability of making $3 is [tex]P(X > 1 ) = 0.67668[/tex]
and the probability of making -$1 is [tex] P(X \le 1 ) = 0.32332[/tex]
Generally the value of profit per day is mathematically represented as
[tex]E(X) = 3 * P(X > 1 ) + (-1 * P(X \le 1 ) )[/tex]
=> [tex]E(X) = 3 * 0.67668 + (-1 * 0.32332 )[/tex]
=> [tex]E(X) = \$ 1.7067 [/tex]
