More about Binomial Coefficients: (handout)
A) (x+y)n has binomial coefficients in the expansion.
nCk xkyn-k.
Use this to show that Binomial distribution satisfies the law
1 = P(0) +P(1) +.....+P(n) (Prob. of everything = 1)
How? Let x = p, y = q. Then 1 = (p + q)n
= P(0) +P(1) +.....+P(n)
B) Pascal's triangle: Interactive Probability: Bernoulli Trials.
Each pair sums to the one below. nCk = (n-1)C(k-1) + (n-1)Ck
Sum of paths argument: number of paths to position (n,k) = sum
of number of paths to 2 positions just above. (For Discrete alums--a
setup to an argument by induction)
Will start here Wednesday.
Geometric distribution: How many
flips to the first head?
Model. Identical Bernoulli trials: Continue till first
Success. (Note, not exactly independent because you quit at
the first head, but probability of success on any single trial stays the
same--is independent of number of trials.)
Y = # of trials. Sample space: (1,2,3,4,..........) A discrete
but not finite sample space. Do tree.
P(Y = k) = (1-p)k-1p = qk-1p (the sequence
FF....FS, k-1 failures followed by 1 success)
Some books use W = # of failures before the first success.
Y = W+1.
Does it work? Do the probabilities sum to 1?
p + qp + q2p + q3p +.......= 1.
See p. 36, Geometric Series, Finite Geometric Series
. . Know these!
"It takes at least k to get a success" Prob = No success on first
k-1. = qk-1
" Success happened on or before k'th" =
p + qp + q2p + q3p + ...+ qk-1p
= Finite Geometric Series .
Negative binomial (p. 52) Same as Geometric, only number
of trials till the r'th success. Formula for k trials: The
last in the string is S; before that are k-1 letters, of which r-1 are
S's (and k-r are F's). So that part is binomial.
P(Y = k) = (k-1)C(r-1) pr-1qk-r p = (k-1)C(r-1)
prqk-r What are the possible
values for k?
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - -
Reading: Ash pp. 36. pp. 51-2. Ahead, Poisson dist, sec
2-5, pp. 63-67
HW:
A. Remember Day 5: We can now prove ( nC1-nC2+nC3-nC4+.....+
nCn = 1). Hint: Let x = 1 and y = 1 and look at (x-y)n
This theorem, for P(A or B
or C....), was used in problem 7d, p. 52, due today.
B. Pascal's triangle: a) Complete the row for n = 8, using
the handout and the row for 7.
b) Use algebra/arithmetic to show nCk = (n-1)C(k-1) + (n-1)Ck.
Try for smallish numbers, n =6, k=4, etc. See if you can show
it in the general case, using algebra.
Rest will be assigned Wednesday:
C. Geometric/Neg. Binomial: a) Give
formula for " Success happened on or before k'th" = p
+ qp + q2p + q3p + ...+ qk-1p =
? Use the Finite Geometric Series . Check that this plus P(no
successes on first k) =1.
b) Use Interactive Probability (Bernoulli Trials:
Experiment: Number of Trials for k successes): to get the histogram for
the Geometric distribution (k=1). Copy, roughly, the distributions
when p =. 2 and p = .8, on the same scale. Note each probability
is q times the previous one.
c) Use Interactive Probability as in b.
With k = 2, run the value for p back and forth (between .2 and 1) to see
the distribution. Repeat with 3, 4, 5. Write what the shape is like
at the extremes and how it changes.
Freund problem sheet:
13, 15, 17
Ash p. 53, 16 (a,b,c are straightforward.
There are several ways of getting d. The shortest is the trickiest.
For e, write down the favorable sequences and their probabilities.
Then you can use the Geometric Series.)
17. Make a tree of the fav's.
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