Chapter 5, Section 9 Multivariate Distributions

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Chapter 5, Section 9 Multivariate Distributions
The Multinomial Probability Distributions
? John J Currano, 03/30/2009
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A multinomial experiment is a generalization of
the binomial from 2 outcomes to k outcomes (e.g.,
toss a k-sided die, not necessarily fair).It has
the following properties

• It consists of n identical trials.
• The outcome of each trial falls into one of k
  classes or cells.
• For 1 i k, the probability, pi, that the
  outcome of the trial falls in class i remains the
  same from trial to trial. Moreover, p1 p2 ?
  ? ? pk 1.
• The trials are independent.
• The random variables of interest are Y1, Y2, . .
  . , Yk, where Yi the number of trials in which
  the outcome falls into cell i. Moreover, Y1 Y2
  ? ? ? Yk n.

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Definition. If Y1, Y2, . . . , Yk are as in the
previous definition, we say that they have a
(joint) multinomial distribution with parameters
n and p1, p2, . . . , pk. Their joint
probability function is
for yi 0, 1, 2, . . . , n and
and p(y1, y2, . . . , yk) 0 elsewhere.
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Proof of the statement about the probability
function ____ ____ ____
____ trial 1 2
3 ? ? ? n The probability of any
string with y1 ones, y2 twos, . . . , and yk ks
(where y1 y2 ? ? ? yk n ) is
since the trials are independent. Since the
number of such strings is
P(Y1 y1, , Yk yk ) is the product of these
two expressions.
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where yi 0, 1, 2, . . . , n and
Note.
is called a multinomial coefficient and is
denoted
By the Multinomial Theorem, it is the coefficient
of
in the expansion of (a1  a2  ? ? ?  ak)n.
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Theorem. If X bin(n, p), then (X, n X)
Multinomial(n, p, 1 p). Theorem. Suppose (Y1,
. . . , Yk ) Multinomial(n, p1, . . . , pk ).
Then the marginal distribution of Yi is bin(n, pi
) for i 1, 2, , k. Proof. Consider the
experiment to result in just two outcomes
success (class i occurs) and failure (some other
class occurs). The properties of a multinomial
experiment guarantee that, looked at in this way,
it is a binomial experiment with Yi successes
bin(n, pi ). Corollary. If (Y1, . . . , Yk )
Multinomial(n, p1, . . . , pk ), then E(Yi ) n
pi and V(Yi ) n pi (1 pi ). Theorem.
If (Y1, . . . , Yk ) Multinomial(n, p1, . . . ,
pk ), then Cov(Ys, Yt ) ?n ps pt , if s ? t.
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Example (p. 283 5.123) Among residential fires,
73 are in family homes, 20 are in apartments,
and 7 are in other types of dwellings. If 4
residential fires are independently reported on a
single day, find the probability that 2 are in
family homes, 1 is in an apartment, and 1 is in
another type of dwelling. Solution. Let Y1
of fires in family homes Y2 of fires in
apartments Y3 of fires in other types of
dwellings. Then (Y1, Y2, Y3) Multinomial(4,
0.73, 0.20, 0.07), and we are asked for p(2, 1,
1)
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Example (p. 283 5.124) Given the situation
above, and that the typical cost of damages
caused by fire is 20,000 for family homes,
10,000 for apartments, and 2,000 for other
types of dwellings, find the expected total
damage cost and the variance of the total damage
cost if 4 fires are independently
reported. Solution. The total damage cost is C
20,000Y1 10,000Y2 2,000Y3 where (Y1, Y2,
Y3) Multinomial(4, 0.73, 0.20, 0.07). Thus,
E(C) E(20,000Y1 10,000Y2 2,000Y3 )
20,000E(Y1 ) 10,000E(Y2 ) 2,000E(Y3 )
20,000 ? 4(0.73) 10,000 ? 4(0.20) 2,000 ?
4(0.07) 58,400 8,000 560 66,960.
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C 20,000Y1 10,000Y2 2,000Y3 (Y1, Y2,
Y3) Multinomial(4, 0.73, 0.20, 0.07) V(C)
V(20,000Y1 10,000Y2 2,000Y3 )
20,0002 V(Y1 ) 10,0002 V(Y2 ) 2,0002 V(Y3
) 2(20,000)(10,000) Cov(Y1, Y2)
2(20,000)(2,000) Cov(Y1, Y3) 2(10,000)(2,000)
Cov(Y2, Y3) 20,0002 ? 4(.73)(.27)
10,0002 ? 4(.20) 0.80) 2,0002 ? 4(.07)(0.93)
2(20,000)(10,000) ?4 (0.73)(0.20)
2(20,000)(2,000) ?4 (0.73)(0.07)
2(10,000)(2,000) ?4 (0.20)(0.07)
128,209,600. ?C 11,322.97.