1 Loi de désintégration radioactive a

T > t T
R+t∈R
t∈RTR+
R+t < 0T > t
R+R
T∼ E(λ)
λ f :R+→R+f(x)=1−λe−λx
P(T≥t) = 1 −P(T < t)
= 1 −P(T≤t)
= 1 −Zt
−∞
f(x)dx
=e−λx
P(T=t) = 0
t7→ Rt
−∞ f(x)dx
λ
IR
P(T∈I) = ZR
I(x)f(x)dx
I(x)=1 x∈I I = [t, +∞)

e−λt
e−N0λt
p N0
pN0
T∼ E(λ)P(T < t) = 1 −e−λt 1−e−λt =1
2
p= 1 −P(T < t1
2)=1/2
t1
2N0/2
p= 1 −P(T < 2t1
2) = e−2λln(2)
λ=1
4N0/4

F(x) = −1
λf(x)
F0=f
τ=E[T] = Z+∞
0
xf(x)dx
= [xF (x)]+∞
0−Z+∞
0
F(x)dx
=1
λZ+∞
0
f(x)dx
=1
λ
fR+R+∞
0f(x)dx = 1
τ=t1
2/ln(2) 1
ln(2) ≈1.44
λ=ln(2)
t1
2s−1
P(T < t) = 1 −e−λt
t= 4 10
h−1
t∗∈R+t≥t∗P(T < t)≥0.95
e−λt∗= 0.95 ⇔t∗=−ln(0.95)
λ