BSc: Probability And Statistics.f22 - Revision history

R.sirgalina: /* Final assessment */

2023-03-23T09:37:59Z

Final assessment

← Older revision		Revision as of 09:37, 23 March 2023
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	=== Final assessment ===		=== Final assessment ===
−	~~====~~ Section 1 ~~====~~	+	'''Section 1'''
	# The probabilities for three students to pass the exam are equal to <math display="inline">\frac{11}{12}</math>, <math display="inline">\frac{11}{14}</math> and <math display="inline">\frac{18}{25}</math> respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.		# The probabilities for three students to pass the exam are equal to <math display="inline">\frac{11}{12}</math>, <math display="inline">\frac{11}{14}</math> and <math display="inline">\frac{18}{25}</math> respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.
	# One of 10-digit numbers in which digits go in non-increasing order is chosen at random. Find the probability that exactly 4 different digits are used in this number.		# One of 10-digit numbers in which digits go in non-increasing order is chosen at random. Find the probability that exactly 4 different digits are used in this number.
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	# Two dice are rolled simultaneously. What is the probability that the sum is even given that it is a multiple of 3?		# Two dice are rolled simultaneously. What is the probability that the sum is even given that it is a multiple of 3?
	# There are 5 white balls and 7 green balls in the first urn; 2 white balls and 10 green balls in the second urn. The third urn, that has initially been empty, is filled with the balls: 4 balls are taken from the first urn, 6 balls are taken from the second urn, and they are placed into the third urn. After that, 2 balls are taken at random from the third urn. It turns out that both these balls are green. Determine the probability that these balls originate from different urns.		# There are 5 white balls and 7 green balls in the first urn; 2 white balls and 10 green balls in the second urn. The third urn, that has initially been empty, is filled with the balls: 4 balls are taken from the first urn, 6 balls are taken from the second urn, and they are placed into the third urn. After that, 2 balls are taken at random from the third urn. It turns out that both these balls are green. Determine the probability that these balls originate from different urns.
−	~~====~~ Section 2 ~~====~~	+	'''Section 2'''
	# Is it possible for random variable <math display="inline">X</math> to have a binomial distribution if (a) <math display="inline">E X=6</math> and <math display="inline">Var X=3</math>; (b) <math display="inline">E X=7</math> and <math display="inline">Var X=4</math>?		# Is it possible for random variable <math display="inline">X</math> to have a binomial distribution if (a) <math display="inline">E X=6</math> and <math display="inline">Var X=3</math>; (b) <math display="inline">E X=7</math> and <math display="inline">Var X=4</math>?
	# Let <math display="inline">Y</math> be number of sixes and <math display="inline">Z</math> be number of fours one gets when rolling six dice. Find the expected value and variance of <math display="inline">Y+Z</math>.		# Let <math display="inline">Y</math> be number of sixes and <math display="inline">Z</math> be number of fours one gets when rolling six dice. Find the expected value and variance of <math display="inline">Y+Z</math>.
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	# Random variable <math display="inline">\theta</math> is exponentially distributed with parameter <math display="inline">\lambda</math>. Calculate the probabilities that <math display="inline">\theta</math> belongs to intervals <math display="inline">(0;1), (1;2), \ldots, (n-1;n), \ldots</math> and show that these probabilities form a geometric sequence. What is the common ratio of this sequence?		# Random variable <math display="inline">\theta</math> is exponentially distributed with parameter <math display="inline">\lambda</math>. Calculate the probabilities that <math display="inline">\theta</math> belongs to intervals <math display="inline">(0;1), (1;2), \ldots, (n-1;n), \ldots</math> and show that these probabilities form a geometric sequence. What is the common ratio of this sequence?
	# It is known that <math display="inline">\xi</math> is normally distributed random variable, and <math display="inline">P\{\|\xi-E\xi\|<1\}=0{.}3</math>. Find the probability that <math display="inline">\|\xi-E\xi\|<2</math>.		# It is known that <math display="inline">\xi</math> is normally distributed random variable, and <math display="inline">P\{\|\xi-E\xi\|<1\}=0{.}3</math>. Find the probability that <math display="inline">\|\xi-E\xi\|<2</math>.
−	~~====~~ Section 3 ~~====~~	+	'''Section 3'''
	# The probabilities for three students to pass the exam are equal to <math display="inline">\frac{11}{12}</math>, <math display="inline">\frac{11}{14}</math> and <math display="inline">\frac{18}{25}</math> respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.		# The probabilities for three students to pass the exam are equal to <math display="inline">\frac{11}{12}</math>, <math display="inline">\frac{11}{14}</math> and <math display="inline">\frac{18}{25}</math> respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.
	# Let us consider independent identically distributed random variables with uniform distribution on <math display="inline">(\theta;\theta+3)</math>. Find the maximum likelihood estimator of <math display="inline">\theta</math>. Which one of these estimators is unbiased? Justify your answer.		# Let us consider independent identically distributed random variables with uniform distribution on <math display="inline">(\theta;\theta+3)</math>. Find the maximum likelihood estimator of <math display="inline">\theta</math>. Which one of these estimators is unbiased? Justify your answer.

R.sirgalina: /* Probability and Statistics */

2023-03-13T08:02:46Z

Probability and Statistics

← Older revision		Revision as of 08:02, 13 March 2023
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	== Short Description ==		== Short Description ==
	The course is designed to provide Software Engineers and Computer Scientists by correct knowledge of basic (core) concepts, definitions, theoretical results and applied methods & techniques of Probability Theory and Mathematical Statistics. The main idea of the course is to study mathematical basis of modelling random experiments. The course includes constructing a probability space, a model of a random experiment, and its applications to practice. After that, random variables and their properties are considered. As examples of applying this theoretical background, limit theorems of probability theory are proved (law of large numbers, central limit theorem) and some elements of mathematical statistics are studied.		The course is designed to provide Software Engineers and Computer Scientists by correct knowledge of basic (core) concepts, definitions, theoretical results and applied methods & techniques of Probability Theory and Mathematical Statistics. The main idea of the course is to study mathematical basis of modelling random experiments. The course includes constructing a probability space, a model of a random experiment, and its applications to practice. After that, random variables and their properties are considered. As examples of applying this theoretical background, limit theorems of probability theory are proved (law of large numbers, central limit theorem) and some elements of mathematical statistics are studied.
		+
		+	== Prerequisites ==
		+
		+	=== Prerequisite subjects ===
		+
		+	=== Prerequisite topics ===

	== Course Topics ==		== Course Topics ==
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	=== Software and tools used within the course ===		=== Software and tools used within the course ===
	* No.		* No.
		+
		+	= Teaching Methodology: Methods, techniques, & activities =

	== Activities and Teaching Methods ==		== Activities and Teaching Methods ==

I.konyukhov at 07:01, 9 September 2022

2022-09-09T07:01:26Z

← Older revision		Revision as of 07:01, 9 September 2022
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	\| Midterm \|\| 20		\| Midterm \|\| 20
	\|-		\|-
−	\| ~~Quizzes~~ \|\| 28 (14 for each)	+	\| Tests \|\| 28 (14 for each)
	\|-		\|-
	\| Final exam \|\| 50		\| Final exam \|\| 50

I.konyukhov at 09:46, 8 September 2022

2022-09-08T09:46:12Z

← Older revision		Revision as of 09:46, 8 September 2022
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	\| Midterm \|\| 20		\| Midterm \|\| 20
	\|-		\|-
−	\| Quizzes \|\| 28 (2 for each)	+	\| Quizzes \|\| 28 (14 for each)
	\|-		\|-
	\| Final exam \|\| 50		\| Final exam \|\| 50

I.konyukhov: /* Final assessment */

2022-06-28T08:17:11Z

Final assessment

← Older revision		Revision as of 08:17, 28 June 2022
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	=== Final assessment ===		=== Final assessment ===
−	~~'''~~Section 1~~'''~~	+	==== Section 1 ====
	# The probabilities for three students to pass the exam are equal to <math display="inline">\frac{11}{12}</math>, <math display="inline">\frac{11}{14}</math> and <math display="inline">\frac{18}{25}</math> respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.		# The probabilities for three students to pass the exam are equal to <math display="inline">\frac{11}{12}</math>, <math display="inline">\frac{11}{14}</math> and <math display="inline">\frac{18}{25}</math> respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.
	# One of 10-digit numbers in which digits go in non-increasing order is chosen at random. Find the probability that exactly 4 different digits are used in this number.		# One of 10-digit numbers in which digits go in non-increasing order is chosen at random. Find the probability that exactly 4 different digits are used in this number.
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	# Two dice are rolled simultaneously. What is the probability that the sum is even given that it is a multiple of 3?		# Two dice are rolled simultaneously. What is the probability that the sum is even given that it is a multiple of 3?
	# There are 5 white balls and 7 green balls in the first urn; 2 white balls and 10 green balls in the second urn. The third urn, that has initially been empty, is filled with the balls: 4 balls are taken from the first urn, 6 balls are taken from the second urn, and they are placed into the third urn. After that, 2 balls are taken at random from the third urn. It turns out that both these balls are green. Determine the probability that these balls originate from different urns.		# There are 5 white balls and 7 green balls in the first urn; 2 white balls and 10 green balls in the second urn. The third urn, that has initially been empty, is filled with the balls: 4 balls are taken from the first urn, 6 balls are taken from the second urn, and they are placed into the third urn. After that, 2 balls are taken at random from the third urn. It turns out that both these balls are green. Determine the probability that these balls originate from different urns.
		⚫	==== Section 2 ====
−
⚫	~~'''~~Section 2~~'''~~
	# Is it possible for random variable <math display="inline">X</math> to have a binomial distribution if (a) <math display="inline">E X=6</math> and <math display="inline">Var X=3</math>; (b) <math display="inline">E X=7</math> and <math display="inline">Var X=4</math>?		# Is it possible for random variable <math display="inline">X</math> to have a binomial distribution if (a) <math display="inline">E X=6</math> and <math display="inline">Var X=3</math>; (b) <math display="inline">E X=7</math> and <math display="inline">Var X=4</math>?
	# Let <math display="inline">Y</math> be number of sixes and <math display="inline">Z</math> be number of fours one gets when rolling six dice. Find the expected value and variance of <math display="inline">Y+Z</math>.		# Let <math display="inline">Y</math> be number of sixes and <math display="inline">Z</math> be number of fours one gets when rolling six dice. Find the expected value and variance of <math display="inline">Y+Z</math>.
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	# Random variable <math display="inline">\theta</math> is exponentially distributed with parameter <math display="inline">\lambda</math>. Calculate the probabilities that <math display="inline">\theta</math> belongs to intervals <math display="inline">(0;1), (1;2), \ldots, (n-1;n), \ldots</math> and show that these probabilities form a geometric sequence. What is the common ratio of this sequence?		# Random variable <math display="inline">\theta</math> is exponentially distributed with parameter <math display="inline">\lambda</math>. Calculate the probabilities that <math display="inline">\theta</math> belongs to intervals <math display="inline">(0;1), (1;2), \ldots, (n-1;n), \ldots</math> and show that these probabilities form a geometric sequence. What is the common ratio of this sequence?
	# It is known that <math display="inline">\xi</math> is normally distributed random variable, and <math display="inline">P\{\|\xi-E\xi\|<1\}=0{.}3</math>. Find the probability that <math display="inline">\|\xi-E\xi\|<2</math>.		# It is known that <math display="inline">\xi</math> is normally distributed random variable, and <math display="inline">P\{\|\xi-E\xi\|<1\}=0{.}3</math>. Find the probability that <math display="inline">\|\xi-E\xi\|<2</math>.
		⚫	==== Section 3 ====
−
⚫	~~'''~~Section 3~~'''~~
	# The probabilities for three students to pass the exam are equal to <math display="inline">\frac{11}{12}</math>, <math display="inline">\frac{11}{14}</math> and <math display="inline">\frac{18}{25}</math> respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.		# The probabilities for three students to pass the exam are equal to <math display="inline">\frac{11}{12}</math>, <math display="inline">\frac{11}{14}</math> and <math display="inline">\frac{18}{25}</math> respectively. Determine the probability that at least one student passes the exam given that they pass or fail independently of each other.
	# Let us consider independent identically distributed random variables with uniform distribution on <math display="inline">(\theta;\theta+3)</math>. Find the maximum likelihood estimator of <math display="inline">\theta</math>. Which one of these estimators is unbiased? Justify your answer.		# Let us consider independent identically distributed random variables with uniform distribution on <math display="inline">(\theta;\theta+3)</math>. Find the maximum likelihood estimator of <math display="inline">\theta</math>. Which one of these estimators is unbiased? Justify your answer.

I.konyukhov: /* Open access resources */

2022-06-23T19:46:02Z

Open access resources

← Older revision		Revision as of 19:46, 23 June 2022
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	=== Open access resources ===		=== Open access resources ===
−	* Probability & statistics for engineers & scientists/Ronald E. Walpole ... [et al.] — 9th ed. p. cm. ISBN 978-0-321-62911-1	+	* Probability & statistics for engineers & scientists/Ronald E. Walpole ... [et al.] — 9th ed. p. cm. ISBN 978-0-321-62911-1 [https://math.buet.ac.bd/public/faculty_profile/files/835598806.pdf book]
−	[https://math.buet.ac.bd/public/faculty_profile/files/835598806.pdf book]
	* Durrett Rick. (2019) Probability. Theory and Examples,		* Durrett Rick. (2019) Probability. Theory and Examples,

I.konyukhov: /* Section 2 */

2022-06-23T19:39:58Z

Section 2

← Older revision		Revision as of 19:39, 23 June 2022
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	# Six people entered the lift at the ground floor of a nine-storied house. Find the expected value for (a) the number of stops where exactly one person gets off the lift; (b) the number of stops where exactly two persons leave the lift.		# Six people entered the lift at the ground floor of a nine-storied house. Find the expected value for (a) the number of stops where exactly one person gets off the lift; (b) the number of stops where exactly two persons leave the lift.
	# Find the expected value and variance of <math display="inline">a^\xi</math> given that <math display="inline">\xi\sim Bin(n,p)</math>.		# Find the expected value and variance of <math display="inline">a^\xi</math> given that <math display="inline">\xi\sim Bin(n,p)</math>.
−	# Random variable <math display="inline">Y</math> has a uniform distribution on interval <math display="inline">(a;b)</math>, and <math display="inline">E Y=\Var Y=3</math>. Find <math display="inline">a</math> and <math display="inline">b</math>.	+	# Random variable <math display="inline">Y</math> has a uniform distribution on interval <math display="inline">(a;b)</math>, and <math display="inline">E Y= Var Y=3</math>. Find <math display="inline">a</math> and <math display="inline">b</math>.

	==== Section 3 ====		==== Section 3 ====

I.konyukhov: /* Final assessment */

2022-06-23T19:38:49Z

Final assessment

@@ Line 228: / Line 228: @@
 '''Section 2'''
 '''Section 3'''
 === The retake exam ===

I.konyukhov: /* Section 3 */

2022-06-23T19:37:33Z

Section 3

← Older revision		Revision as of 19:37, 23 June 2022
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	==== Section 3 ====		==== Section 3 ====
	# How many times does one have to flip a coin to get the results “heads”, “heads” in succession? Is the result going to change if we replace the sequence with “tails”, “heads”?		# How many times does one have to flip a coin to get the results “heads”, “heads” in succession? Is the result going to change if we replace the sequence with “tails”, “heads”?
−	# Forty three equally strong sportsmen take part in a ski race; 18 of them belong to club ~~{\textstyle~~ A~~}{\textstyle A}~~, 10 to club ~~{\textstyle~~ B~~}{\textstyle B}~~ and 15 to club ~~{\textstyle~~ C~~}{\textstyle C}~~. What is the average place for (a) the best participant from club ~~{\textstyle~~ B~~}{\textstyle B}~~; (b) the worst participant from club ~~{\textstyle B}{\textstyle~~ B}?	+	# Forty three equally strong sportsmen take part in a ski race; 18 of them belong to club <math display="inline">A</math>, 10 to club <math display="inline">B</math> and 15 to club <math display="inline">C</math>. What is the average place for (a) the best participant from club <math display="inline">B</math>; (b) the worst participant from club <math display="inline">B</math>?
	# How much rolls does one need on average to get a sequence “6”, “6” when rolling a symmetric six-sided die? And if we change this sequence to “6”, “6”, “6”?		# How much rolls does one need on average to get a sequence “6”, “6” when rolling a symmetric six-sided die? And if we change this sequence to “6”, “6”, “6”?
−	~~{\textstyle~~ ~~\zeta }{\textstyle~~ \zeta } is the quantity of threes and ~~{\textstyle~~ \eta ~~}{\textstyle \eta }~~ is the quantity of odd digits obtained when rolling a fair die ~~{\textstyle K}{\textstyle~~ K} times. Find correlation coefficient between ~~{\textstyle~~ \eta ~~}{\textstyle \eta }~~ and ~~{\textstyle~~ \zeta ~~}{\textstyle \zeta }~~.	+	# <math display="inline">\zeta</math> is the quantity of threes and <math display="inline">\eta</math> is the quantity of odd digits obtained when rolling a fair die <math display="inline">K</math> times. Find correlation coefficient between <math display="inline">\eta</math> and <math display="inline">\zeta</math>.

	=== Final assessment ===		=== Final assessment ===

I.konyukhov: /* Section 3 */

2022-06-23T19:36:48Z

Section 3

← Older revision		Revision as of 19:36, 23 June 2022
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	==== Section 3 ====		==== Section 3 ====
−	№ How many times does one have to flip a coin to get the results “heads”, “heads” in succession? Is the result going to change if we replace the sequence with “tails”, “heads”?	+	# How many times does one have to flip a coin to get the results “heads”, “heads” in succession? Is the result going to change if we replace the sequence with “tails”, “heads”?
−	№ Forty three equally strong sportsmen take part in a ski race; 18 of them belong to club {\textstyle A}{\textstyle A}, 10 to club {\textstyle B}{\textstyle B} and 15 to club {\textstyle C}{\textstyle C}. What is the average place for (a) the best participant from club {\textstyle B}{\textstyle B}; (b) the worst participant from club {\textstyle B}{\textstyle B}?	+	# Forty three equally strong sportsmen take part in a ski race; 18 of them belong to club {\textstyle A}{\textstyle A}, 10 to club {\textstyle B}{\textstyle B} and 15 to club {\textstyle C}{\textstyle C}. What is the average place for (a) the best participant from club {\textstyle B}{\textstyle B}; (b) the worst participant from club {\textstyle B}{\textstyle B}?
−	№ How much rolls does one need on average to get a sequence “6”, “6” when rolling a symmetric six-sided die? And if we change this sequence to “6”, “6”, “6”?	+	# How much rolls does one need on average to get a sequence “6”, “6” when rolling a symmetric six-sided die? And if we change this sequence to “6”, “6”, “6”?
	{\textstyle \zeta }{\textstyle \zeta } is the quantity of threes and {\textstyle \eta }{\textstyle \eta } is the quantity of odd digits obtained when rolling a fair die {\textstyle K}{\textstyle K} times. Find correlation coefficient between {\textstyle \eta }{\textstyle \eta } and {\textstyle \zeta }{\textstyle \zeta }.		{\textstyle \zeta }{\textstyle \zeta } is the quantity of threes and {\textstyle \eta }{\textstyle \eta } is the quantity of odd digits obtained when rolling a fair die {\textstyle K}{\textstyle K} times. Find correlation coefficient between {\textstyle \eta }{\textstyle \eta } and {\textstyle \zeta }{\textstyle \zeta }.