Using F(x) to find percentiles
("quantiles"). (Not in text)
Def: xp is the
p-th percentile if 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(x) = p is something we can solve
for x ).
Graphically,
find p on the y-axis, go over to the F curve and down to xp.
Quantile
Applet, choose CDF, any distribution.
Start here Friday: (I may add more to this
outline on Friday's page)
4-3: Exponential distribution
Binomial: n trials, probability
p
of success on a single trial.
Waiting time X till first success:
Geometric distribution.
P(X= 6) = qqqqqp, 5 trials with NO successes, before the success.
E(X) = 1/p
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 (calculate
this later)
Finding distribution of Exponential: Find F(x).
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.
Find that probability by using the Poisson distribution
.
Let W = # of arrivals in the interval (0,
xo ). The average is lambda per unit.
Since the length of the interval for W is x
o, the parameter for W is (lambda x
o)
No successes in an interval of length xo,
Poisson prob. is exp(- lambda x o)
= P(X > xo).
F(x) = 1- exp(- lambda x).
f(x) = F'(x) = lambda exp(- lambda x).
(Sketch: = 0 for negative
x. Lambda at x=0. downward curving from there)
Gamma
Applet k=1, r = lambda. Run and watch the Poisson process happening.
(Vertical scale on left is in tenths, but goes above the
max lambda. Hard to read.)
Like the Geometric, it's memoryless: Start anywhere, it's
as if there was no past. (Because in the Poisson, arrivals are equally
likely everywhere)
If you have waited for time to already, the probability
of having to wait at least x more is the same as just having
to wait at least x, starting from 0:
P( X > x+to |X > to) = P(X > x).
Proof (like the Geometric). (p. 124)
Another application: Lifetime of something
that has no "aging"--just subject to random "death". lambda = 1/average
lifetime.
- - - - - - -- - - - - - - - - - - - - - - -
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. 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.) Gamma
Applet , k is for waiting till k'th arrival. In Quantile
Applet, you can see the Gamma generalization: k is now the "shape"
parameter, and can take on values between the integers.)
- - - - - - -- - - - - - - - - - - - - - - -
HW B and
C added
A. On the Density sheet ( f(x) = x-.5,
1<x<2) 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.)
B. Use the Quantile
Applet, different distributions and parameters, to get used to these
ideas. For the Standard Normal distribution, find the first and 3rd
quartiles (25th and 75th percentiles.) Also the four quintiles (20,
40, 60, 80%iles)--government economic data is mostly in quintiles.
C. For (Ash p.106) the Uniform distribution
on the interval [a,b], find the 25th percentile by graphical or geometric
methods. Now find it by solving 1/4 = F(x) = (x-a)/(b-a) for x. Check
your answer. Now solve p = F(x) = (x-a)/(b-a) for x, getting xp,
as a formula for the p'th percentile in a uniform distribution.
HW postponed (read
4.3, including Gamma) Ash p. 127
All Exponential except #8:
#1.
#2 (for e, you can use your Poisson table.)
#3e (cf. the bird calls sheet), #4
#5
#6
#7 [part f should say 4 particles--my book says
3. My book also has 210 instead of 10 in the answer to g]
#8 Gamma
B. Try out the applets, get used to the shapes and concepts...
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