EXERCICE 1

Donnees pour moins de 25 ans (n=100)
Classes x     (0,5)  (5,15)  (15,25)
fi            19.5   43.2    37.3
FCC           19.5   62.7   100.0
ci             2.5    10      20

1) Mediane Me
   classe mediane = (5,15) car FCC passe de 19.5 a 62.7
   Me = 5 + (15-5)*(50-19.5)/(62.7-19.5)
      = 5 + 10*30.5/43.2
      = 5 + 7.0648
      = 12.0648
   Me approx 12.1

2) Moyenne xbar
   numerator = 2.5*19.5 + 10*43.2 + 20*37.3
             = 48.75 + 432 + 746
             = 1226.75
   xbar = 1226.75/100
        = 12.2675
   xbar approx 12.27

3) Ecart-type sigma
   m2 = 2.5^2*19.5 + 10^2*43.2 + 20^2*37.3
      = 6.25*19.5 + 100*43.2 + 400*37.3
      = 121.875 + 4320 + 14920
      = 19361.875
   Var = m2/100 - xbar^2
       = 193.61875 - 150.5106
       = 43.10815
   sigma = sqrt(43.10815)
         = 6.566
   sigma approx 6.6

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EXERCICE 2

n=1000, p=0.2
Xi ~ Bernoulli(0.2) i.i.d.
X = sum(Xi) ~ Binomial(1000,0.2)

1) E(X) = n*p = 1000*0.2 = 200
   Var(X) = n*p*(1-p) = 1000*0.2*0.8 = 160

2) Par TCL X ~ Normal(200,160)
   P(X>=250) approx 1 - Phi((249.5-200)/sqrt(160))
                 = 1 - Phi(49.5/12.649)
                 = 1 - Phi(3.913)
                 ~ 0.000045

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EXERCICE 3

N=1200, p=0.2
Xi ~ Bernoulli(0.2)
X = sum(Xi) ~ Binomial(1200,0.2)

a) E(X) = 1200*0.2 = 240
   Var(X)= 1200*0.2*0.8 = 192

b) Par TCL X ~ Normal(240,192)
   P(X>=250) approx 1 - Phi((249.5-240)/sqrt(192))
                 = 1 - Phi(9.5/13.856)
                 = 1 - Phi(0.686)
                 = 1 - 0.754
                 = 0.246

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EXERCICE 4

n=100, phat = 0.15

a) Xi ~ Bernoulli(p)
   phat = sum(Xi)/100

b) IC 95% pour p :
   phat +/- 1.96*sqrt(phat*(1-phat)/100)
   = 0.15 +/- 1.96*sqrt(0.15*0.85/100)
   = 0.15 +/- 1.96*0.03571
   = 0.15 +/- 0.06996
   = [0.080 , 0.220]

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EXERCICE 5

Xi iid ~ Exponential(lambda), n>=2

1) lnL = n*ln(lambda) - lambda*sum(Xi)
   d(lnL)/dlambda = n/lambda - sum(Xi) = 0
   => lambda_hat = n / sum(Xi)

2) E(lambda_hat) = (n-1)/n * lambda
   biais = -lambda/n => non sans biais
   asymp sans biais car (n-1)/n->1

3) Var(lambda_hat)->0 => converge vers lambda

4) lambda_tilde = (n/(n-1))*lambda_hat
   E(lambda_tilde)=lambda et converge vers lambda