Programme de Khôlle n 9 (semaine 47) Probabilités

n◦
N N∗
N
5
0,1
•N
•X
•Y
N=X+Y
n k p(N=n)(X=k)
k p(X=k)X
0,5
Y4,5 (i, j)∈N2
p((X=i)∩(Y=j))
X Y
500 300
36
10−3
1000 X
X

n
p= 0,1n0,1
p(N=n)(X=k) = n
k0,1k0,9n−k
n < k p(N=n)(X=k) = 0 {N=n}n
p(X=k) =
+∞
X
n=k
p((N=n)∩(X=k)) =
+∞
X
n=k
p(N=n)(X=k)×p(N=n)
=
+∞
X
n=k
e−55n
n!×n k0,1k0,9n−k=e−50,1k
+∞
X
n=k
n!5n
n!k!(n−k)!0,9n−k
=e−50,1k5k
k!
+∞
X
n=k
5n−k0,9n−k
(n−k)! =e−50,5k k!
+∞
X
K=0
4,5K
K!
| {z }
e4,5
=e−0,50,5k
k!.
X0,5
p((X=i)∩(Y=j)) = p((X=i)∩(N=i+j)) = p(N=i+j)×p(N=i+j)(X=i)
=e−55i+j
(i+j)! i+j i0,1i0,9j=e−55i+j×0,1i0,9j i!j!=e−5×0,5i4,5j i!j!.
Y
4,5
p(X=i)×p(Y=j) = e−0,50,5i
i!×e−4,54,5j j!
1
500 36 N
36 n= 300 p=1
500 = 0,05
p(N= 2) = 500
20,0520,95298 '0.
p(N≥2) = 1 −p(N= 0) −p(N= 1) '1
X
n= 1000 p= 10−3
B(\,√)' P(\√)np = 1

p(X= 2) 'e−112
2! '1 2e
p(X≥2) = 1 −p(X < 2) = 1 −p(X= 0) −p(X= 1) '1−e−1−e−1'e−2
e