Places to go from here: Multinomial distribution. Urn problems (cf. Hypergeometric). Binomial Coefficients. Geometric Distribution. Poisson Distribution.
Multinomial Distribution: Still n
independent, identical trials,
Generalize from Two outcomes to 3, (4, etc.) outcomes. 4 outcomes
A,
B, C, D. Prob's P(A)....P(D)
Suppose n trials. Each possible
result is an n-string of A, B, C, D.
Let x, y, z, w be the number of each kind of outcome in the result.
x + y +z + w = n
Any particular string has prob P(A)xP(B)yP(C)zP(D)w
How many different strings? (Ash p. 45-6)
Multinomial coefficient (Derivation like Binomial) # of
rearrangements of x A's, y B's, z C's, w D's.
Take the n letters, label them so all distinguishable. A1
A2 A3 B1 B2 C1
C2 D. n! rearrangements.
How many look alike without the labels? The A's can be rearranged
among themselves 3! = x! different ways , The B's 2! = y! different ways,
etc.
Each rearrangement of A's can go with any
rearrangement of B's, with any rearrangement of C's, etc.
So for any given list, e.g. ABBACADC, there are x!y!z!w! different
versions if you can distinguish the A's, the B's etc.
(One is A3B1B2 A2C1A1DC2
)
Multinomial coefficient = n! / (x!y!z!w!)
Urn problems (Ash pp. 48-50): N balls. Labeled S, F.
(or A, B, C, D)
Drawing n with replacement:
= Binomial/Multinomial because the urn is the same each time.
Drawing n without replacement:
Removing each ball changes the contents of the urn by 1.
Tree model has subsequent probabilities changing correspondingly.
Easier to calculate using committees/poker&bridge
hand techniques.
2 kinds of balls = Hypergeometric Dist. (D defectives)
More kinds of balls: Like poker/bridge hands.
Limit-result: If N is large in proportion to n,
and the number of balls of each kind drawn is small in proportion to the
number in the urn, then drawing without replacement may be approximated
by drawing with replacement.
Interactive probability:
"Urn" (Binomial/Multinomial only. Red = Success=Defective)
Model: compare
Sampling with replacement and Sampling without replacement.
- - - - - - - - - - - - - - - - - - -
Reading: Ash pp. 46-51, M&M 375-9, 387-9 (Ash postpones
means and s.d.'s. Do a little review in M&M to remember what
the words mean, for a distribution.)
HW:
A) Urn problem: N = 12 balls, 4 Red. Draw n =3.
1) Without replacement: Calculate the
distribution values, for x = 0, 1, 2, 3 Reds.
Do it two ways: Hypergeometric formula, and tree with 3 branchings.
2) With replacement: Calculate the distribution
values, for x = 0, 1, 2, 3 Reds.
Do it two ways: Binomial formula, and tree with 3 branchings.
3) Go to the Interactive Probability: Urn program
and check your answers. Also record the mean and s.d. for both distributions.
On the same graph, graph the two bar graphs showing the distribution.
Which is more spread out, the with or without replacement version?
4) Still in Interactive Probability, increase
the number of balls, but keep the proportion. N = 99, R = 33. Draw
n = 3.
Record the distributions and mean and s.d., with and without replacement.
Make the two trees, and label the branches. Note similarity/difference.
Ash: pp. 52ff. For these, write out the formulas for the
results. Calculate if it's not too hard; or, if suitable, stick it
into Interactive Probability.
1, 2, 4, 5, 8, 9, 12
7 (This is Multinomial, with 5 outcomes. See if you can figure
out how to see it that way, before looking at the answer.)
6
Freund Handout 7, 11
| Sievers home | Math300-Sp02/Day12.htm | 11pm | 2/21/02 |