Exercicio de estatistica

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Pedro Henrique Bargado de Sousa 52940
MGST1

Grupo I

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X U unif (0,1) ∩ X ~ U N,F (2,5)

1-(1/(b-a)) + 3 * (1/(b-a)) = 1 (=) 1/(b-a) + 1/(b-a) = 1 (=) (b-a) = 4(X)

Fx (X) = ⌠ f(x) x dx

x< 0 Fx (X) = ⌠ 0 dx = 0

0≤ x ≤ 1 Fx (X) = ⌠ f(x) dx = ⌠ 0 dx + ⌠ 1/4 dx =( 1/4)x

1≤ x ≤ 2 Fx (X) = ⌠ f(x) dx = ⌠ 0 dx + ⌠ 1/4 dx + ⌠ 0 dx = 1/4

2≤ x ≤ 5 Fx (X) = ⌠ f(x) dx = ⌠ 0 dx + ⌠ 1/4 dx + ⌠ 0 dx + ⌠ 1/4 dx = = 1/4 – ((x-2)/4)

1/4 se 0≤ x ≤ 1 V 2 ≤ x ≤ 5 f(x) 0 se

0 se x < 0

(x/4) se 0≤ x ≤ 1

F(x) 1/4 se 1 < x < 2

(1/4) + ((x-2)/4) se 2 ≤ x ≤ 5

1 se x > 5

b)

E (X) = ⌠ f(x) x dx = ⌠ (1/4) x dx + ⌠ (1/4) x dx = 1/4 [( x^2) / 2] + 1/4 [( x^2) / 2] = = 11/4

Var (X) = E ( X^2 ) – (E(X)^2)

(E(X)^2) = 121/16

E ( X^2 ) = ⌠ f(x) x^2 dx = ⌠ (1/4) x^2 dx + ⌠ (1/4) x ^2 dx = 1/4 [( x^3) / 3] + 1/4 [( x^3) / 3] = (1/4) * (1/3) + (1/4) * ((5^3 - 2^3)/3)= 1/12 + 1/4* ((126-8)/3) = 1/12 + 117/12 = 59/6
Var (X) = 59/6 – 121/16 = 2.27

Grupo II

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