Student Notes To Accompany MS4214: STATISTICAL INFERENCE

Example 2.6 (Radioactive Decay). In this classic set of data Rutherford and Geiger
counted the number of scintillations in 72 second intervals caused by radioactive decay
of a quantity of the element polonium. Altogether there were 10097 scintillations during
2608 such intervals:
Count 0 1 2 3 4 5 6 7
Observed 57 203 383 525 532 408 573 139
Count 8 9 10 11 12 13 14
Observed 45 27 10 4 1 0 1
The Poisson probability mass function with mean parameter θ is
The likelihood function equals
fX(x|θ) = θx exp (−θ)
.
x!
L(θ) = � θ xi exp (−θ)
xi!
The relevant mathematical calculations are
= θ� xi exp (−nθ)
� .
xi!
ℓ(θ) = (Σxi) ln (θ) − nθ − ln [Π(xi!)]
S(θ) =
⇒ ˆ θ =
� xi
θ
� xi
n
− n
= ¯x
I(θ) = Σxi
> 0,
θ2 ∀ θ
implying ˆ θ is MLE. Also E( ˆ θ) = � E(xi) = 1 �
θ = θ, so θˆ is an unbiased estimator.
� Var(xi) = 1
Next Var( ˆ θ) = 1
n2 nθ and I(θ) = E[I(θ)] = n/θ = (Var[ˆ θ]) −1 implying
that ˆ θ attains the theoretical CRLB. It is always useful to compare the fitted values
from a model against the observed values.
i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Oi 57 203 383 525 532 408 573 139 45 27 10 4 1 0 1
Ei 54 211 407 525 508 393 254 140 68 29 11 4 1 0 0
n
+3 -8 -24 0 +24 +15 +19 -1 -23 -2 -1 0 -1 +1 +1
The Poisson law agrees with the observed variation within about one-twentieth of its
range. �
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