Homework: You should learn what we have taken in class and read
and study the following chapters
2 Outcomes and events 10
3 Relative frequency and probability 12
5 Classical problems 16
from http://www.math.bme.hu/~vetier/df/Part-I.pdf Problem to solve:Use the
table
Y = the number of grandparents
4
0,05
0,12
0,1
0,06
3
0,05
0,08
0,12
0,04
2
0,02
0,06
0,06
0,02
1
0,01
0,04
0,05
0,02
0
0,01
0,04
0,03
0,02
0
1
2
3
X = the number of children
(available also in 2016-02-17__05_Children_and_grandparents.xlsx) to answer the following questons:What is
the probability that
1. the number of children is equal to 3 ?
Answer: 0,02 + 0,02 + 0,02 + 0,04
+ 0,06 = 0,16
2. the number of grandparents is equal to
2?
Answer: 0,02 + 0,06 + 0,06
+ 0,02 = 0,16 3. the number of children is equal to
the number of grandparents?
Answer: 0,01 + 0,04 + 0,06
+ 0,04 = 0,15
4. the number of children is greater than the number of
grandparents? Answer:
0,04 + 0,03 + 0,02 + 0,05 + 0,02 + 0,02 =
0,18 ----------------------------------------------- 2016-02-22 Monday 2016-02-22__01_Children_and_grandparents_SOLUTION.xlsx 2016-02-22__02_Lottery.xlsx 2016-02-22__03_Hypergeometrical_distribution.xlsx 2016-02-22__04_Fixing_references_by_F4_key.xlsx 2016-02-22__05_Hypergeom_AGAIN.xlsx Homework:
1. Use the file 2016-02-22__03_Hypergeometrical_distribution.xlsx
to tell how much is the probability that the number of boys in the
selected group is 4 or less. 2. Construct a file similar to 2016-02-22__03_Hypergeometrical_distribution.xlsx
so that the total number of students is 400 and the number of boys
is 150, and the size of the group is 77.
3. (Continued) How much is the probability that the number of boys
in the selected group of size 77 is
a) exactly equal to 40 ?
b) less than or equal to 40 ? You may send your solution in an attached file to
vetier@math.bme.hu with the
subject ProbHW .
No credits are given for the solution but you know: "Practice
makes the master". -----------------------------------------------
Homework: 1. LEARN EVERYTHING WE TOOK IN CLASS!!! You may
use the "Photo pictures of the board" above. 2. Read and learn chapters
1 Introductory
problems
4 Random numbers
6 Geometrical
problems, uniform distributions
from
http://www.math.bme.hu/~vetier/df/Part-I.pdf
3. Solve:
My friend goes to work by bus and metro every
day. He waits for the bus no more than 10 minutes.
The amount of time he waits for the bus is
uniformly distributed between 0 and 10.
When he changes to the metro, he waits
for the metro no more than 5 minutes.
No matter how much he waited for the bus,
the amount of time he waits for the metro is uniformly distributed
between 0 and 5 . What is the probability
that 1. the waiting
time for the metro is less than the waiting time for the bus ?
2. the waiting time for the
metro is less than 2 times the waiting time for the bus ? 3. the
waiting time for the metro plus the waiting time for the bus is
less than 7 minutes?
4. the waiting time for the
metro plus 2 times the waiting time for the busis less
than 7 minutes?
5. the waiting time for the
metro plus the waiting time for the bus is less than z minutes,
where z is a number
5.a) between 0 and 5 ?
5.b) between 5 and 10 ?
5.c) between 10 and 15 ?
Give your answer to each of these questions in terms of the
variable z .
Hint: First figure out the sample
space for the two-dimensional random variable ( X , Y ) .
Then figure out the set of all favorable outcomes for each event.
Calculate the areas and then take the corresponding ratios. ----------------------------------------------- 2016-02-29 Monday 2016-02-29__01_Bus_and_metro.xlsx
Homework: 1. LEARN EVERYTHING WE TOOK IN CLASS! You
may use the "Photo pictures of the board" above. 2. Read and learn chapters 1
Introductory problems 3
2 Outcomes and events 10
3 Relative frequency and probability 12
4 Random numbers 14
5 Classical problems 16
6 Geometrical problems, uniform
distributions 19
7 Basic properties of probability
22
8 Conditional relative frequency
and conditional probability
from
http://www.math.bme.hu/~vetier/df/Part-I.pdf
3. Solve:
1. In a certain city, 30 percent of the people are Conservatives,
50 percent are Liberals,
and 20 percent
are Independents. Records show that in a particular election
65 percent of the
Conservatives voted, 82 percent of the Liberals voted, and 50
percent of the
Independents voted. If a person in the city is selected at random
and it is learnt that
he did not vote in the last election, what is the probability that
he is a Liberal?
2. There are two boxes: a red and a blue. In
the red box, there are 3 red and 2 blue
balls. In the blue box,
there are 3 red and 7 blue balls. First, we pick a ball from
the red box, and put it
into the blue box. Then we pick a second ball from the
blue box, and put it
into the red box. Finally, we pick a third ball from the red
box again.
(a) What is the probability that the first ball is red and the
second is blue?
(b) What is the probability that the second ball is blue?
(c) What is the probability that the first ball is red, the second
is blue and the
third ball is red again?
(d) What is the probability that the second is blue and the third
is red?
(e) What is the probability that both the first and the second
balls are red?
(f) What is the probability that both the second ball is red?
(g) What is the probability that all the three balls are red?
(h) What is the probability that both the second and the third
balls are red?
(i) What is the probability that the third ball is red?
(j) On condition that the third ball is red, what is the
probability that the second
ball is red, too?
(k) On condition that the third ball is red, what is the
probability that the second
ball is red, but the first was blue?
(l) On condition that the third and second balls were red, what is
the probability
that the first is blue?
Next class: This Saturday (as if
it were a Monday) at 8:15!
----------------------------------------------- 2016-03-05
Saturday 2016-03-05__01_Total_prob_formula_Bayes_formula__application.xlsx Photo
pictures of the board: Homework: Is he sick or
healthy?
Assume that 0.001 part of people are infected by a certain bad
illness, 0.999 part of people are healthy. Assume also that if a
person is infected by the illness, then he or she will be
correctly diagnosed sick with a probability 0.9, and he or she
will be mistakenly diagnosed healthy with a probability 0.1.
Moreover, if a person is healthy, then he or she will be correctly
diagnosed healthy with a probability 0.8. and he or she will be
mistakenly diagnosed sick with a probability 0.2, Now imagine that
a person is examined, and the test says the person is sick.
Knowing this fact what is the probability that this person is
really sick? ----------------------------------------------- 2016-03-07 Monday 2016-03-07__01_Multiplication_rule_for_independent_events.xlsx
----------------------------------------------- 2016-03-09
Wednesday 2016-03-09__01_Binomial_distribution.xlsx
Photo picture of the board: Deriving the binomial
distribution Homework: 1. LEARN EVERYTHING WE TOOK IN CLASS! You may
use the "Photo pictures of the board" above.
Study, understand the meaning of, be able to construct the
file: 2016-03-09__01_Binomial_distribution.xlsx
2. Read and learn chapter 1-4 from http://www.math.bme.hu/~vetier/df/Part-II.pdf
1 Discrete random
variables and distributions
2 Uniform
distribution (discrete)
3
Hyper-geometrical distribution
4 Binomial
distribution 3.
Solve:
1. Blue eyed girls. Assume that
3/4 of girls in a country have blue eyes. If you choose 20 girls
at
random in that country, then what is the probability that
(a) exactly 15 of them have blue eyes;
(b) exactly 16 of them have blue eyes;
(c) exactly 17 of them have blue eyes;
(d) more than 17 of them have blue eyes;
(e) less than 17 of them have blue eyes;
(f) the number of blue eyed girls is between 15 and 17 (equality
permitted)?
2. Teachers becoming sick. There are 70
teachers in our institute. Each teacher, independently of the
others,
may become sick during a day with probability 0.04. What is the
probability that
k of them become sick during a day? Make a table and a figure -
using binomial
distribution - so that k runs from 0 to 20. ----------------------------------------------- 2016-03-16 Wednesday 2016-03_16_HW_Liyangyuan.xlsx 2016-03-16__01_Binomial_distribution.xlsx 2016-03-16__02_Binomial_distribution_HOW_MANY_CHAIRS.xlsx 2016-03-16__03_Binomial_distribution__AIRPLANE_TICKETS.xlsx -------------------- First
Midterm Test:
21 March,
Monday during
our lesson
Material of the test:
Chapters
1-9 from http://www.math.bme.hu/~vetier/df/Part-I.pdf
Chapters
1-4 from http://www.math.bme.hu/~vetier/df/Part-II.pdf
Homework:
1. Learn
Chapter
15 from
http://www.math.bme.hu/~vetier/df/Part-II.pdf
2. Make
a simulation
file to
simulate a
random
variable with
negative
binomial
distribution. 3.
Solve:
1. Asking for
help on a
highway When
your car breaks down on a highway and you ask ask for help. Assume
that each driver, independently of the other stops and helps you
with a probability 0.2.
What is the probability that
(a) exactly,
(b) at most,
(c) at least
5 cars pass without giving you help before somebody will help you?
2. Number of
injured people Assume
that when a 5 passenger car has an accident, then the number X of
injured people, independently of any other factors, has the
following distribution:
P(X = 0) = 0.4,
P(X = 1) = 0.2,
P(X = 2) = 0.1,
P(X = 3) = 0.1,
P(X = 4) = 0.1,
P(X = 5) = 0.1,
and when an 8 passenger bus has an accident, then the number Y of
injured people, independently of any other factors, has the
following distribution:
P(Y = 0) = 0.50,
P(Y = 1) = 0.10,
P(Y = 2) = 0.10,
P(Y = 3) = 0.05,
P(Y = 4) = 0.05,
P(Y = 5) = 0.05,
P(Y = 6) = 0.05,
P(Y = 7) = 0.05,
P(Y = 8) = 0.05.
(a) How much is the expected value of the number of injured people
when a 5 passenger car has an accident?
(b) How much is the expected value of the number of injured people
when an 8 passenger bus has an accident?
(c) How much is the expected value of the number of injured people
when a 5 passenger car hits an 8 passenger bus?
----------------------------------------------- First
REPEATED
Midterm Test:
6
April,
Wednesday 6
pm, K234
Material of
the test is
the same:
Chapters
1-9 from
http://www.math.bme.hu/~vetier/df/Part-I.pdf
Chapters
1-4 from
http://www.math.bme.hu/~vetier/df/Part-II.pdf ----------------------------------------------- 2016-04-04 Monday 2016-04-04__01_averege_of_geometrical_experiments.xlsx 2016-04-04__02_averege_of_binomial_experiments.xlsx 2016-04-04__03_averege_of_Poisson_experiments.xlsx 2016-04-04__03_averege_of_squares_of_X.xlsx 2016-04-04__04_averege_of_3rd_powers_of_X.xlsx 2016-04-04__05_averege_of_binomial_experiments_squared.xlsx 2016-04-04__06_HLOOKUP_COMMAND.xlsx
---------------------------------------------
Learn
from
Part-IV:
5
Normal distributions in two-dimensions
12 Limit theorems to normal distributions and study the
Excel simulation files in it.
Solve:
Let
us consider the two-dimensional random variable (X, Y ), which
follows normal
distribution with parameters mu1 = 26, sigma1 = 4,
mu2 = 14, sigma2 = 2, r = 0.6.
How much is the probability that 12 < Y < 14 on condition
that
(a) X = 25?
(b) X = 26?
(c) X = 27?
-------------------------------------
Rep.
Test 2
results:
B4S695
19
JSK3WO
11
N0G7VR
7
RUNSYG
13
X8TTOF
12
XZ9HGG
12
-----------------------------------------------
2016-05-23 Monday IMG_20160523_084011.jpg IMG_20160523_084121.jpg IMG_20160523_084415.jpg IMG_20160523_084633.jpg IMG_20160523_085109.jpg IMG_20160523_085336.jpg 2016-05-23___01.xlsx
sample
exam:
exam_2015_12_14.pdf
-------------------------------
Extra repeated
test: 25 May,
Wednesday,
12:00, EIB
Registration
in NEPTUN is
needed.
Only
one of the two
tests can be
repeated ------------------------------ Exams:
1, 8 15 of
June
(Wednesdays),
8:15.
Place:
H601 Material of the exams: The
following
chapters from
http://www.math.bme.hu/~vetier/df/Part-I.pdf
1 Introductory problems
2 Outcomes and events
3 Relative frequency and probability
4 Random numbers
5 Classical problems
6 Geometrical problems, uniform
distributions
7 Basic properties of
probability
8 Conditional relative
frequency and conditional probability The
following
chapters from
http://www.math.bme.hu/~vetier/df/Part-II.pdf
5 Geometrical distribution (pessimistic)
6 Geometrical distribution (optimistic)
7 Negative binomial distribution (pessimistic)
8 Negative binomial distribution (optimistic)
9 Poisson-distribution
15 Expected value of discrete distributions
16 Expected values of the most important discrete
distributions
17 Expected value of a function of a discrete
random variable
18 Moments of a discrete random variable The
following
chapters from
http://www.math.bme.hu/~vetier/df/Part-III.pdf
1 Continuous random variables
2 Distribution function
4 Density function
6 Uniform distributions
7 Distributions of some functions of random numbers
8 Arc-sine distribution
10 Beta distributions
11 Exponential distribution
13 Normal distributions
16 Expected value of continuous distributions
17 Expected value of a function of a continuous
random variable
19 Standard deviation, etc. The
following
chapters from
http://www.math.bme.hu/~vetier/df/Part-IV.pdf
1 Two-dimensional
random variables and distributions
2 Uniform distribution on a two-dimensional set
3 Beta distributions in two-dimensions
4 Projections and
conditional distributions
5 Normal distributions in two-dimensions
6 Independence of random variables
8 Properties of the expected value, variance and
standard deviation
12 Limit theorems to normal distributions - Moivre-Laplace
theorem Form of the exam: written exam, 6
problems, each consisting of two parts, each worth 10 points.
Types of questions:
Prorblem 1: Elementary probability or a one- or
two-dimensional problem with discrete "distributions without
name"
Prorblem 2: One or two dimensional problem with discrete
"distributions with name" (like binomial, Poisson ...)
Prorblem 3: One-dimensional continuous distribution
Prorblem 4: Two-dimensional continuous distribution
Prorblem 5: Topic we took after the second test
Prorblem 6: Simulation (Answer will be given probably on paper,
since no computer will be available)
One of the two parts in three of the problems will be a
"theoretical question" (defining or describing some notions,
declaring or proving some theorems ...)
This means that theory makes one fourth of the points.
---------------------------------------
Whole term results:
TEST1
TEST2
what
what
Total
Neptun
Test1
Rep
Reprep
counts
Test2
Rep
RepRep
counts
Signature
points
B4S695
0
8
8
No
EF3JIT
No
GBW8MN
No
JSK3WO
8
10
10
2
11
11
Gets
21
N0G7VR
4
0
0
7
7
No
QAG82U
16
16
12
12
Gets
28
RUNSYG
4
12
12
4
13
13
Gets
25
S11MNE
3
3
3
No
VE8OT9
13
13
No
X8TTOF
12
12
8
12
12
Gets
24
XDL7MS
11
11
10
10
Gets
21
XZ9HGG
2
11
11
0
12
12
Gets
23
2016-06-01 Exam results:
midterm
Exam
Total
points
points
points
Grade
EF3JIT
*
22
48
70
3
JSK3WO
*
21
30
51
2
QAG82U
*
28
37
65
3
RUNSYG
*
25
0
25
1
VE8OT9
*
22,5
30
52,5
2
X8TTOF
*
24
21
45
1
XDL7MS
*
21
41
62
3
XZ9HGG
*
23
10
33
1
Students may come and see their exams and/or ask questions:
2016-06-07 Tuesday, 10am, H502
-----
Nobody has registered for the June 8 exam, so it
is cancelled.
Next exam (This is the last one!): June 15, 8:15, H601.
Do not forget to register in time in Neptun.
