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Different - Probability with Physical Applications - Exam, Exams of Probability and Statistics

AMET UniversityProbability and Statistics

This is the Exam of Probability with Physical Applications which includes Probability, Different Pairs, Complete Pairs, Exactly, Least One Right, Shoe are Chosen, Certain Coin, Probability, Approximations etc. Key important points are: Different, First Two Places, License Plates, Possible, Terms, Black and White, Selected, Police Department, Station, Reserve

Typology: Exams

2012/2013

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Be sure this exam has 10 pages including the cover
The University of British Columbia
Sessional Exams – 2005 Term 2
Mathematics 318 Probability with Physical Applications, All sections
Dr. D. Brydges, Dr. G. Slade
Name:
Student Number:
This exam consists of 8questions worth 10 marks each. No aids other then calculators are
permitted.
Problem total possible score
1. 10
2. 10
3. 10
4. 10
5. 10
6. 10
7. 10
8. 10
total 80
1. Each candidate should b e prepared to produce his library/AMS card upon request.
2. Read and observe the following rules:
No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave
during the first half hour of the examination.
Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities
in examination questions.
CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the
examination and shall be liable to disciplinary action.
(a) Making use of any books, papers or memoranda, other than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness
shall not be received.
3. Smoking is not permitted during examinations.
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Different - Probability with Physical Applications - Exam and more Exams Probability and Statistics in PDF only on Docsity!

Be sure this exam has 10 pages including the cover

The University of British Columbia

Sessional Exams – 2005 Term 2

Mathematics 318 Probability with Physical Applications, All sections

Dr. D. Brydges, Dr. G. Slade

Name:

Student Number:

This exam consists of 8 questions worth 10 marks each. No aids other then calculators are

permitted.

Problem total possible score

total 80

  1. Each candidate should be prepared to produce his library/AMS card upon request.
  2. Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers or memoranda, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received.
  3. Smoking is not permitted during examinations.

Tables on last page.

(2 points) 1. (a)How many different 7-place license plates are possible if the first two places are for

letters and the other 5 are for digits?

(3 points) (b)Explain, in terms of black and white balls selected from an urn, why

∑^ n

j=

n

j

2 n

n

(3 points) (c)A police department consists of 10 officers. If the policy is to have 5 officers on patrol in

the streets, 2 full time at the station and 3 on reserve at the station then how many

divisions of the 10 officers into the three groups are possible?

(2 points) (d)How many different ways can 22 people divide themselves up to play soccer (11 per side).

(10 points) 3. Ten numbers are rounded to the nearest integer and then summed. Using the central limit

theorem, determine the probability that the sum of the rounded numbers will equal the

rounded sum of the unrounded numbers. You may assume that the roundoffs for the ten

numbers are independent and uniformly distributed in (− 0. 5 , 0 .5).

4. Earthquakes happen according to a Poisson process with rate λ but each quake is detected

with probability p, independently. Let X denote the number of earthquakes in a fixed unit

time interval and let Xc denote the number that are detected in the same time interval.

(2 points) (a)Write a formula for the moment generating function of a discrete random variable Y in

terms of the probability mass distribution of Y.

(2 points) (b)What is the moment generating function of X. Explain, using part (a).

(1 points) (c)What is the distribution of Xc conditioned on X = n?

(2 points) (d)Calculate E(etXc^ |X = n).

(2 points) (e)What is the moment generating function of Xc?

(1 points) (f)What is the distribution of Xc?

6. Consider the Markov chain with state space { 0 , 1 , 2 , 3 , 4 , 5 } and transition matrix

P =

(3 points) (a)Draw the transition diagram showing the six states with arrows indicating possible

transitions and their probabilities.

(4 points) (b)Determine all the irreducibility classes of this Markov chain.

(3 points) (c)Determine which states are recurrent and which are transient.

7. This problem considers a modification of the Ehrenfest chain. Let M be a positive integer.

Suppose that M molecules are distributed among two urns. We choose a molecule at random

and remove it from its urn, and then choose an urn at random and place the removed

molecule into the chosen urn. Let Xn denote the number of molecules in urn number one,

after the nth^ step. This defines a Markov chain.

(4 points) (a)Write formulas for the transition matrix elements Pij.

(3 points) (b)Guess the stationary distribution for general M.

(3 points) (c)Verify that your guess in (c) is correct.

Table 1: Mean and Variances

Distribution Mean Variance Bin (n, p) np np(1 − p) Geometric (p) 1 p^1 p− 2 p Poisson (λ) λ λ

 - (b−a) Uniform (a, b) a+ 2 b - Exp (λ) λ^1 λ
  • 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0. Table 2: cdf of normal distribution
  • 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.
  • 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.
  • 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.
  • 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.
  • 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.
  • 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.
  • 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.
  • 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.
  • 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.
  • 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.
  • 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.
  • 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.
  • 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.
  • 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.
  • 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.
  • 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.
  • 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.
  • 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.
  • 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.
  • 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.
  • 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.
  • 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.
  • 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.
  • 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.
  • 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.
  • 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.
  • 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.
  • 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.
  • 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.
  • 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.
  • 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.