Math 300 , Spring 2002, Day30, F, April 12 Hit
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Class
Mini-exam on the continuous material, F, f
etc, through percentiles with F. Open book, papers. No collaboration.
No questions on the exam to Mary--ask me any questions. Handed out
today, due next Friday.
Homework questions?
4-3: Exponential distribution
Poisson: A unit interval, with independent
"arrivals" . Y = # of arrivals in the unit interval. lambda
= E(Y) = parameter.
Waiting time X till first
success: Exponential distribution.
E(X) = 1/lambda, Var = 1/(lambda)2 (calculate soon)
F(x) = 1- exp(- lambda x).
f(x) = F'(x) = lambda exp(- lambda x).
(sketch)
Like the Geometric, it's memoryless: P( X > x+to
|X > to) = P(X > x). Proof, done.
Another application: Lifetime of something
that has no "aging"--just subject to random "death". lambda = 1/average
lifetime.
(I didn't say this in class)
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Gamma distribution with parameter n:
analogue of Negative Binomial: Waiting time till n'th arrival.
Strategy for finding distribution.
Find F(xo) = P(X < x o) = 1 - P(X
> xo).
P(X > xo) is the probability
that in the interval (0, xo), there are NO arrivals,
Or 1, Or 2,.....Or n-1. (the n'th is yet to come.)
Each of those
is a Poisson probability. Sum them. When we take the derivative
to get f(x), all terms cancel but one.
f(x)
= lambda times Poisson probability for n-1 arrivals, parameter (lambda
x).
See p. 125 for the computation.
(Why "Gamma"? The gamma function
is a generalization of the factorial k! to values of k that are not integers.
The Gamma distribution has a factorial in the denominator. It can
be generalized to use non-integer n as parameter (but doesn't have the
interpretation of waiting till n'th arrival any more.))
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Using F(x) to find (theoretical) percentiles.
(Not
in Ash)
Let xp be the p-th percentile,
that is P(X
< xp ) = p
For example, the median x0.5
is the number such that P(X < x0.5 ) = 0.5, the number
with probability 1/2 below it.
Since F(x) = P(X < x),
F(xp ) = P(X
< xp ) = p
So to find xp , we can set F(x)
= p and solve for x (assuming F is a formula we can solve for
x in).
Graphically,
find p on the y-axis, go over to the F curve and down to xp.
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HW Ash p. 127
A. Show that the formula for the exponential
distribution "is" the formula for the gamma, when n =1 (that is, the exponential
is a special case of the gamma.)
#6 c,d gamma
#8 gamma
B. On the Density sheet (with the graphs)
use the graphical method (over and down) to find, approximately, the median.
The 80th percentile. Then solve the equation F(xp
) = p for the x's for p = .5 and .8. (Check the two methods
give approximately the same answers.)
Exponential handout: Data file in Mac lab,
on NT machine, or download Excel file
Mini-exam covers to here.
C. Review integration by parts.
Do this standard example: integral from 0 to 1 of f(x) = xex.
----- ---- ---- ----
Monday, start here:
Expected values and variance: (back
to ch. 7)
Instead of sums, we have integral signs.
E(X) is still the balance point on the f(x)
graph.
"Law of the unconscious statistician"--
If Y = g(X), then E(Y) = integral over all y's
of y times density of Y, OR
= integral over all x's of g(x) times
density of X.
So E(X2) is the integral
of x2 times f(x).
Var(X) = E(X2) -µ2
still. (proof)
Find E(X) for exponential (p. 214). Requires
integration by parts (or Ash's handy formula p. 94).
HW:
Expected value and variance, back to Ch.
7 pp. 213=125, 225-232. Useful integrals p. 95
Find E(X) and Var(X) for the Density sheet formula
(f = x - 1/2, 1<x<2).
Ash p. 221,
#1,
#2 Do the calculus, get the obvious answer.
#1, #3 (use p. 95)
#15
there will be more.
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