STAT3613 Tutorial 02

(b) b = - ( the value of X at the point of inflection ) × c (c) c = 4 × ( slope at the point of inflection ) /a (d) d = minimum value of Y Solutions Since 0 ≤ 1 1 + e - ( b + cX ) ≤ 1 ⇒ d ≤ a 1 + e - ( b + cX ) + d ≤ a + d, therefore, min( Y ) = d, max( Y ) = a + d = ⇒ a = max Y - min Y, d = min( Y ) . At the point of infection, Y 00 = 0 , Y 0 = ace - ( b + cX ) (1 + e - ( b + cX ) ) - 2 Y 00 = ac 2 h 2 e - 2( b + cX ) (1 + e - ( b + cX ) ) - 3 - e - ( b + cX ) (1 + e - ( b + cX ) ) - 2 i = ac 2 e - ( b + cX ) (1 + e - ( b + cX ) ) - 2 h 2 e - ( b + cX ) (1 + e - ( b + cX ) ) - 1 - 1 i = ac 2 e - ( b + cX ) (1 + e - ( b + cX ) ) - 2 2 e b + cX + 1 - 1 At the point of infection, ( X = ˜ x , Y 0 = ˜ y 0 ), 0 = 2 e b + c ˜ x + 1 - 1 , = ⇒ e b + c ˜ x = 1 . Thus, b + c ˜ x = 0 = ⇒ b = - c ˜ x, and ˜ y 0 = ac (1) - 1 ( 1 + (1) - 1 ) - 2 = ac 4 = ⇒ c = 4˜ y 0 a Logistic Function: 3. For electronic company B, salesforce is to be allocated to 3 different products, the digital camera (DC), the mobile phone (MP) and the laptop (LP). The current sales for DC, MP and laptop are 5000, 10000 and 4000 respectively. The current salespeople allocated to DC are 25 and that for MP are 35, while that for laptop are 20. The average cost for each salesperson is $40. The margin of DC is $0.8 and that of MP and laptop are $0.7 and $0.9. Historical data of the sales (in %) for various numbers of salespeople (in %) are shown in the table. 3