Corrigé du 30 P. 474

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a. \[\begin{aligned} \operatorname E(X) &= p = 0,74.& \\ \operatorname V(X) &= p(1-p) = 0,74\times 0,26 = 0,1924.& \end{aligned}\] On en déduit que : \[\begin{aligned} \operatorname E(M) &= \operatorname E(X) = 0,74.& \\ \operatorname V(M) &=\frac 1 {80}\operatorname V(X) = \frac{0,1924}{80} = 0,002405.& \\ \sigma(M) &= \sqrt{\operatorname V(M)} = \sqrt{0,002405} \approx 0,049041.& \end{aligned}\]

b. \[\begin{aligned} \operatorname E(X) &= np = 35\times 0,26 = 9,1.& \\ \operatorname V(X) &= np(1-p) = 35\times 0,26 \times 0,74 = 6,734.& \end{aligned}\] Donc: \[\begin{aligned} \operatorname E(M) &= \operatorname E(X) = 9,1.& \\ \operatorname V(M) &=\frac 1 {80}\operatorname V(X) = \frac{6,734}{80} = 0,084175.& \\ \sigma(M) &= \sqrt{\operatorname V(M)} = \sqrt{0,084175} \approx 0,290120.& \end{aligned}\]

c. \[\begin{aligned} \operatorname E(X) &= \frac 1 6 \times 2 + \cdots + \frac 1 6 \times 7 = 4,5.& \\ \operatorname V(X) &=\frac 1 6(2 - 4,5)^2 + \cdots + \frac 16(7-4,5)^2 = \frac{35}{12}.& \end{aligned}\] Donc: \[\begin{aligned} \operatorname E(M) &= E(X) = 4,5.& \\ \operatorname V(M) &= \frac 1{80}\operatorname V(X) = \frac 1{80}\times\frac{35}{12} = \frac 7{192}.& \\ \sigma(M) &= \sqrt{\operatorname V(M)} = \sqrt{\frac{7}{192}} =\frac{\sqrt{21}}{24}.& \end{aligned}\]

d. \[\begin{aligned} \operatorname E(X) &= 0,27\times 0 + \cdots + 0,13\times 3 = 1,14.& \\ \operatorname V(X)&=0,27(0-1,14)^2 + \cdots + 0,13(3-1,14)^2 = 0,9204.& \end{aligned}\] Donc: \[\begin{aligned} \operatorname E(M) &=\operatorname E(X) = 1,14.& \\ \operatorname V(M) &=\frac 1 n \operatorname V(X) = \frac{0,9204}{80} = 0,11505.& \\ \sigma(M) &= \sqrt{\operatorname V(M)} = \sqrt{0,11505} \approx 0,339190.& \end{aligned}\]

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