Group problems 10/31 - PowerPoint PPT Presentation

1 / 14
About This Presentation
Title:

Group problems 10/31

Description:

1. so 2. Compute 3. (a) so (b) Group problems, cont. (c) First Now assume it holds for k=n, so First step analysis yields so Cosmic radiation High-energy particles ... – PowerPoint PPT presentation

Number of Views:59
Avg rating:3.0/5.0
Slides: 15
Provided by: PeterG217
Learn more at: https://stat.uw.edu
Category:

less

Transcript and Presenter's Notes

Title: Group problems 10/31


1
Group problems 10/31
  • 1. so
  • 2. Compute
  • 3. (a)
  • so
  • (b)

2
Group problems, cont.
  • (c) First
  • Now assume it holds for kn, so
  • First step analysis yields
  • so

3
Cosmic radiation
  • High-energy particles continuously bombard the
    atmosphere from all directions
  • Soft part are energetic electrons and gamma ray
    photons
  • When passing through the atmosphere they collide
    with atmospheric atoms creating showers of
    particles
  • Photon hitting an atom produces a pair of
    particles (positronelectron) and loses all its
    energy
  • Electron absorbed in nucleus of atom produces
    photon

4
Furrys model
  • Disregard photons. One electron traveling a
    distance ?t converts into two electrons with
    probability proportional to ?t.
  • P1,n(t) probability of n electrons at depth t
    (starting with one at the top of the atmosphere)

5
  • Rewrite to get
  • Manipulating and letting we get
  • Since P1n(0)1(n1) we solve recursively

6
The (Yule-)Furry process
  • Also called a linear birth process.
  • Problems
  • Absorption of electrons is disregarded
  • Photons are disregarded
  • Energy decrease is neglected
  • Observed shower particles have a maximum at
    about 16 km.
  • particles ?
  • E( particles) ?

7
The general birth process
  • Probability of birth in (t, t?t) when n
    individuals present ?n?t o(?t). X(0)1
  • so

8
Laplace transforms
9
Some examples
10
Back to the birth process
  • Taking Laplace transforms we get
  • which we solve recursively to get
  • Interpretation?

11
A picture
X(t)
?2Exp(?2)
time
?1Exp(?1)
12
A result
  • so by taking Laplace transforms we get
  • When we see that
  • iff the product diverges to zero, i.e.,
  • iff
  • Theorem For any tgt0

13
Example
  • Linear birth process has ?n n???so
  • A birth process with ?n?n2 has
  • A process for which ?P1n(t) lt 1 is called
    dishonest.
  • There is positive probability that the process is
    not in the state space at time t!
  • It has gone off to infinity, or exploded.

14
The Poisson process
  • Birth process with ?n ???X(0)0.
  • What can we say about its construction?
  • What is P(Xtk)?
  • Integration by parts yields
Write a Comment
User Comments (0)
About PowerShow.com