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Answer:
(a) E(Y) = 4400
sd (Y) =225
(b) P(Y ≤ 4500) = 0.67003
(c) P (X₁ > X₂) = 0.31744
Step-by-step explanation:
(a) Here we have
Y = 2·X₁ + 3·X₂
Therefore E(Y) = 2·E(X₁) + 3·E(X₂) = 2000 + 2400 = 4400
sd(Y) is given by
Variance Y = (sd (Y))² = 2²·(sd (X₁))² + 3²·(sd (X₂))²
= 4·8100 + 9·2025 = 50625
sd (Y) = √50625 = 225
(b) The probability that the revenue does not exceed 4500 is given by
P(Y ≤ 4500) = P(z ≤0.444)
z =
z = = 0.444
Therefore from the normal distribution table, we have
P = 0.67003
(c) The probability that the P(X₁ > X₂)
Since the gas station sells 2 portions of X₁ to 3 portions of X₂
Therefore, the probability that the gas station sells more of X₁ is given by
₅C₀ × 2/5⁰×3/5⁵ = 0.07776
₅C₁ × 2/5¹×3/5⁴= 0.2592
₅C₂ × 2/5²×3/5³ = 0.3456
P (X₁ > X₂) = 1 - 0.68256 = 0.31744