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Probability 2 - Notes 6

The Trinomial Distribution
Consider a sequence of n independent trials of an experiment
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If X counts the number of successes, then X ∼ Binomial(n, p)
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If
we write 1 for “success”, 0 for “failure”, and −1 for “neither”, then the outcome of n trials can
be described as a sequence of n numbers
ω = (i1 , i2 ,
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Definition
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The joint distribution of the pare (X,Y ) is called the trinomial distribution
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Theorem
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m
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for (X,Y ) is given by
fX,Y (k, l) = P(X = k,Y = l) =

n!
pk θl (1 − p − θ)n−k−l ,
k!l!(n − k − l)!

where k, l ≥ 0 and k + l ≤ n
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The sample space consists of all sequences of length n described above
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n!
There are n n−k = k!l!(n−k−l)! different sequences with k “successes” (1’s) and l “failures”
k
l
(0’s)
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k!l!(n−k−l)! p θ (1 − p − θ)

2

The name of the distribution comes from the trinomial expansion
(a + b + c)n = (a + (b + c))n =

n

∑

k=0
n n−k

=

∑∑

k=0 l=0

n
k

n k
a (b + c)n−k
k

n n−k
n − k k l n−k−l
n!
ak bl cn−k−l
a bc
=∑∑
l
k!l!(n − k − l)!
k=0 l=0

Properties of the trinomial distribution
1) The marginal distributions of X and Y are just X ∼ Binomial(n, p) and Y ∼ Binomial(n, θ)
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Similar argument works for Y
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Proof
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, (n − y)
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2

This is intuitively obvious
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There
are (n − l) such trials, for each of which the probability that 1 occurs is actually the conditional
p
probability of 1 given that 0 has not occurred, i
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1−θ
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3) We shall now use the results on conditional distributions (Notes 5) and the above properties
to find Cov(X,Y ) and the coefficient of correlation ρ(X,Y )
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According to property 2),
p
p
E[X|Y = l] = (n − l) 1−θ and thus E[X|Y ] = (n −Y ) 1−θ
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In
this case it is easily seen that ρ(X,Y ) = −1
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Denote the
outcomes A1 , A2 ,
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, pk where ∑k p j = 1
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Then

P(X1 = x1 ,
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pk−1 pk j=1
1 2
k−1
x1 !x2 !
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, xk−1 are non-negative integers with ∑k−1 x j ≤ n

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