Download Important properties of Poisson random variables

Important properties of Poisson random variables
Addition of two Poisson random variables (Example 4.5) Let X
and Y be independent Poisson random variables with parameters λ and µ
respectively. Then fX (s) = eλ(s−1) and fY (s) = eµ(s−1) , from Example 4.1.
Then Theorem 4.6 tells us that
fX+Y (s) = eλ(s−1) eµ(s−1) = e(λ+µ)(s−1) .
Using Theorem 4.2 above, we can now conclude that X + Y is a Poisson
random variable with mean λ + µ.
Thinning of a Poisson random variable (Example 6.1) Let N be a
Poisson random variable with mean λ, and let {Xn ; n ∈ N} be a sequence
of independent (of each other and of N ) Bernoulli random variables with
parameter p, i.e. P (Xn = 1) = p and P (Xn = 0) = 1 − p. Now consider
S=
N
X
Xn .
n=1
We can think of this as taking a Poisson distributed number of objects, each
of which has some property with probability p, and selecting only those which
have this property.
The p.g.f. of N is fN (s) = eλ(s−1) by Example 4.1 and the p.g.f. of each Xn
is 1 − p + ps. So using Lemma 6.1, the p.g.f. of S is
eλ(1−p+ps−1) = eλp(s−1)
which we recognise as the p.g.f. of a Poisson random variable with mean λp.
We recall (Theorem 4.2) that the p.g.f. determines the distribution, so we
have shown that the thinned Poisson random variable S also has a Poisson
distribution.
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