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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
Please Subscribe to this Channel for more solutions and lectures
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
History behind the
development of the concept
In 1654, a gambler Chevalier De Metre approached the well known Mathematician Blaise Pascal for certain dice problem. Pascal became interested in these problems and discussed it further with Pierre de Fermat. Both of them solved these problems independently. Since then this concept gained limelight.
Basic Things About The Concept
Probability is used to quantify an attitude of mind towards some uncertain proposition.
The higher the probability of an event, the more certain we are that the event will occur.
It gives detail description about probability, types of probability, difference between mutually exclusive events and independent events, difference between conditional and unconditional probability and Bayes' theorem
This presentation was prepared as part of the curriculum studies for CSCI-659 Topics in Artificial Intelligence Course - Machine Learning in Computational Linguistics.
It was prepared under guidance of Prof. Sandra Kubler.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.2: Addition Rule and Multiplication Rule
History behind the
development of the concept
In 1654, a gambler Chevalier De Metre approached the well known Mathematician Blaise Pascal for certain dice problem. Pascal became interested in these problems and discussed it further with Pierre de Fermat. Both of them solved these problems independently. Since then this concept gained limelight.
Basic Things About The Concept
Probability is used to quantify an attitude of mind towards some uncertain proposition.
The higher the probability of an event, the more certain we are that the event will occur.
It gives detail description about probability, types of probability, difference between mutually exclusive events and independent events, difference between conditional and unconditional probability and Bayes' theorem
This presentation was prepared as part of the curriculum studies for CSCI-659 Topics in Artificial Intelligence Course - Machine Learning in Computational Linguistics.
It was prepared under guidance of Prof. Sandra Kubler.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.2: Addition Rule and Multiplication Rule
Experiment
Event
Sample Space
Unions and Intersections
Mutually Exclusive Events
Rule of Multiplication
Rule of Permutation
Rule of Combination
PROBABILITY
probability and its functions with purpose in the world's situation .pptxJamesAlvaradoManligu
probability and its functions with different context in understanding the purpose of probability in the daily life. it includes the type of probabilities and the functions. Additionally, the components and identities of the importance of relation to the concept of the world. sampling just like random, systematic, stratified sampling.
Adjusting primitives for graph : SHORT REPORT / NOTESSubhajit Sahu
Graph algorithms, like PageRank Compressed Sparse Row (CSR) is an adjacency-list based graph representation that is
Multiply with different modes (map)
1. Performance of sequential execution based vs OpenMP based vector multiply.
2. Comparing various launch configs for CUDA based vector multiply.
Sum with different storage types (reduce)
1. Performance of vector element sum using float vs bfloat16 as the storage type.
Sum with different modes (reduce)
1. Performance of sequential execution based vs OpenMP based vector element sum.
2. Performance of memcpy vs in-place based CUDA based vector element sum.
3. Comparing various launch configs for CUDA based vector element sum (memcpy).
4. Comparing various launch configs for CUDA based vector element sum (in-place).
Sum with in-place strategies of CUDA mode (reduce)
1. Comparing various launch configs for CUDA based vector element sum (in-place).
As Europe's leading economic powerhouse and the fourth-largest hashtag#economy globally, Germany stands at the forefront of innovation and industrial might. Renowned for its precision engineering and high-tech sectors, Germany's economic structure is heavily supported by a robust service industry, accounting for approximately 68% of its GDP. This economic clout and strategic geopolitical stance position Germany as a focal point in the global cyber threat landscape.
In the face of escalating global tensions, particularly those emanating from geopolitical disputes with nations like hashtag#Russia and hashtag#China, hashtag#Germany has witnessed a significant uptick in targeted cyber operations. Our analysis indicates a marked increase in hashtag#cyberattack sophistication aimed at critical infrastructure and key industrial sectors. These attacks range from ransomware campaigns to hashtag#AdvancedPersistentThreats (hashtag#APTs), threatening national security and business integrity.
🔑 Key findings include:
🔍 Increased frequency and complexity of cyber threats.
🔍 Escalation of state-sponsored and criminally motivated cyber operations.
🔍 Active dark web exchanges of malicious tools and tactics.
Our comprehensive report delves into these challenges, using a blend of open-source and proprietary data collection techniques. By monitoring activity on critical networks and analyzing attack patterns, our team provides a detailed overview of the threats facing German entities.
This report aims to equip stakeholders across public and private sectors with the knowledge to enhance their defensive strategies, reduce exposure to cyber risks, and reinforce Germany's resilience against cyber threats.
Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
StarCompliance is a leading firm specializing in the recovery of stolen cryptocurrency. Our comprehensive services are designed to assist individuals and organizations in navigating the complex process of fraud reporting, investigation, and fund recovery. We combine cutting-edge technology with expert legal support to provide a robust solution for victims of crypto theft.
Our Services Include:
Reporting to Tracking Authorities:
We immediately notify all relevant centralized exchanges (CEX), decentralized exchanges (DEX), and wallet providers about the stolen cryptocurrency. This ensures that the stolen assets are flagged as scam transactions, making it impossible for the thief to use them.
Assistance with Filing Police Reports:
We guide you through the process of filing a valid police report. Our support team provides detailed instructions on which police department to contact and helps you complete the necessary paperwork within the critical 72-hour window.
Launching the Refund Process:
Our team of experienced lawyers can initiate lawsuits on your behalf and represent you in various jurisdictions around the world. They work diligently to recover your stolen funds and ensure that justice is served.
At StarCompliance, we understand the urgency and stress involved in dealing with cryptocurrency theft. Our dedicated team works quickly and efficiently to provide you with the support and expertise needed to recover your assets. Trust us to be your partner in navigating the complexities of the crypto world and safeguarding your investments.
Opendatabay - Open Data Marketplace.pptxOpendatabay
Opendatabay.com unlocks the power of data for everyone. Open Data Marketplace fosters a collaborative hub for data enthusiasts to explore, share, and contribute to a vast collection of datasets.
First ever open hub for data enthusiasts to collaborate and innovate. A platform to explore, share, and contribute to a vast collection of datasets. Through robust quality control and innovative technologies like blockchain verification, opendatabay ensures the authenticity and reliability of datasets, empowering users to make data-driven decisions with confidence. Leverage cutting-edge AI technologies to enhance the data exploration, analysis, and discovery experience.
From intelligent search and recommendations to automated data productisation and quotation, Opendatabay AI-driven features streamline the data workflow. Finding the data you need shouldn't be a complex. Opendatabay simplifies the data acquisition process with an intuitive interface and robust search tools. Effortlessly explore, discover, and access the data you need, allowing you to focus on extracting valuable insights. Opendatabay breaks new ground with a dedicated, AI-generated, synthetic datasets.
Leverage these privacy-preserving datasets for training and testing AI models without compromising sensitive information. Opendatabay prioritizes transparency by providing detailed metadata, provenance information, and usage guidelines for each dataset, ensuring users have a comprehensive understanding of the data they're working with. By leveraging a powerful combination of distributed ledger technology and rigorous third-party audits Opendatabay ensures the authenticity and reliability of every dataset. Security is at the core of Opendatabay. Marketplace implements stringent security measures, including encryption, access controls, and regular vulnerability assessments, to safeguard your data and protect your privacy.
Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
Levelwise PageRank with Loop-Based Dead End Handling Strategy : SHORT REPORT ...Subhajit Sahu
Abstract — Levelwise PageRank is an alternative method of PageRank computation which decomposes the input graph into a directed acyclic block-graph of strongly connected components, and processes them in topological order, one level at a time. This enables calculation for ranks in a distributed fashion without per-iteration communication, unlike the standard method where all vertices are processed in each iteration. It however comes with a precondition of the absence of dead ends in the input graph. Here, the native non-distributed performance of Levelwise PageRank was compared against Monolithic PageRank on a CPU as well as a GPU. To ensure a fair comparison, Monolithic PageRank was also performed on a graph where vertices were split by components. Results indicate that Levelwise PageRank is about as fast as Monolithic PageRank on the CPU, but quite a bit slower on the GPU. Slowdown on the GPU is likely caused by a large submission of small workloads, and expected to be non-issue when the computation is performed on massive graphs.
2. Machine Learning
Machine Learning is an interdisciplinary field in Data Science that uses
• statistics
• probability
• algorithms
to learn from data and provide insights which can be used to build
intelligent applications.
2
8. Probability for Data Science
•Probability deals with predicting the likelihood of
future events, while statistics involves the
analysis of the frequency of past events.
8
9. Terminologies
• Event
• Random Variable
• Empirical Probability
• Theoretical Probability
• Joint Probability
• Conditional Probability
9
10. Event
• An event is a set of outcomes of an experiment to which a probability
is assigned.
• E represents event
• P(E) is the probability that the event E occur.
• A situation where E might happen (success) or might not happen
(failure) is called a trial.
10
14. Random Variable
• The variable that represents the outcome of an events is called a
random variable.
• Eg. Getting head or tail in tossing a coin
14
15. Random variable in tossing a coin
• If we toss a coin, the chances for getting head or tail is 50-50
• The probability of getting head or tail is ½ or 50%
• Random variable range between 0 and 1
15
16. Empirical Probability
• Also known as practical probability
• It is the number of times the event occurs divided by the total
number of incidents observed.
• If for ‘n’ trials and we observe ‘s’ successes, the probability of success
is s/n.
• Toss a coin 4 times. The outcome is H, H, H, T
• P(Head) =3/4=0.75
• P(Tail)=1/4=0.25
16
17. Theoretical probability
• The number of ways the particular event can occur divided by the
total number of possible outcomes.
• A head can occur once and possible outcomes are two (head, tail).
The true (theoretical) probability of a head is 1/2.
17
18. Exercise 1
A die is rolled, find the probability that an even number is obtained.
18
19. Exercise 1
A die is rolled, find the probability that an even number is obtained.
Solution:
Let us first write the sample space S of the experiment.
S = {1,2,3,4,5,6}
Let E be the event "an even number is obtained" and write it down.
E = {2,4,6}
We now use the formula of the classical probability.
P(E) = n(E) / n(S) = 3 / 6 = 1 / 2
19
20. Exercise 2
Two coins are tossed, find the probability that two heads are obtained.
Note: Each coin has two possible outcomes H (heads) and T (Tails).
20
21. Exercise 2
Two coins are tossed, find the probability that two heads are obtained.
Note: Each coin has two possible outcomes H (heads) and T (Tails).
The sample space S is given by.
S = {(H,T),(H,H),(T,H),(T,T)}
Let E be the event "two heads are obtained".
E = {(H,H)}
We use the formula of the classical probability.
P(E) = n(E) / n(S) = 1 / 4
21
22. Exercise 3
A card is drawn at random from a deck of cards. Find the probability of
getting the 3 of diamond.
22
23. Exercise 3
A card is drawn at random from a deck of cards. Find the probability of
getting the 3 of diamond.
The sample space S of the experiment in question 6 is shown below
23
24. Exercise 3
A card is drawn at random from a deck of cards. Find the probability of
getting the 3 of diamond.
24
25. Exercise 3
A card is drawn at random from a deck of cards. Find the probability of
getting the 3 of diamond.
Let E be the event "getting the 3 of diamond". An examination of the
sample space shows that there is one "3 of diamond" so that n(E) = 1
and n(S) = 52. Hence the probability of event E occurring is given by
P(E) = 1 / 52
25
26. Exercise 4
The blood groups of 200 people is distributed as follows:
50 have type A blood,
65 have B blood type,
70 have O blood type and
15 have type AB blood.
If a person from this group is selected at random, what is the
probability that this person has O blood type?
26
27. Exercise 4
We construct a table of frequencies for the the blood groups as follows
group frequency
A 50
B 65
O 70
AB 15
We use the empirical formula of the probability
P(E) = Frequency for O blood / Total frequencies
= 70 / 200 = 0.35
27
28. Classwork 1
What is the probability of throwing one dice and getting the number
greater than 4 ?
28
29. Classwork 2
The customer wants to buy a bread and a can. There are 30 pieces of
bread in the shop, including 5 from the previous day, and 20 cans with
unreadable expiration date, of which one has expired. What is the
probability that the customer will buy a fresh bread and a tin under
warranty ?
29
30. Classwork 3
What is the probability that if we choose a trinity from 19 boys and 12
girls, we will have :
a) three boys
b) three girls
c) two boys and one girl ?
30
31. Joint Probability
• Probability of events A and B denoted by P(A and B) or P(A ∩ B) is the
probability that events A and B both occur.
• P(A ∩ B) = P(A). P(B)
• This only applies if A and B are independent, which means that if A
occurred, that doesn’t change the probability of B, and vice versa.
31
32. Conditional Probability
• A and B are not independent
• When A and B are not independent, it is often useful to compute the
conditional probability, P (A|B)
• The probability of A given that B occurred: P(A|B) =
P(A ∩ B)
P(B)
• Similarly, P(B|A) =
P(A ∩ B)
P(A)
32
39. Mutually Exclusive Events
• If two events are NOT independent, then we say that they are dependent.
• Sampling may be done with replacement or without replacement.
• With replacement: If each member of a population is replaced after it is
picked, then that member has the possibility of being chosen more than
once. When sampling is done with replacement, then events are
considered to be independent, meaning the result of the first pick will not
change the probabilities for the second pick.
• Without replacement: When sampling is done without replacement, each
member of a population may be chosen only once. In this case, the
probabilities for the second pick are affected by the result of the first pick.
The events are considered to be dependent or not independent.
39
40. Sampling with replacement
• Suppose you pick three cards with replacement. The first card you
pick out of the 52 cards is the
• Q of spades. You put this card back, reshuffle the cards and pick a
second card from the 52-card deck. It is the ten of clubs. You put this
card back, reshuffle the cards and pick a third card from the 52-card
deck. This time, the card is the Q of spades again. Your picks are {Q of
spades, ten of clubs, Q of spades}. You have picked the Q of spades
twice. You pick each card from the 52-card deck.
40
41. Sampling without replacement
• Suppose you pick three cards without replacement. The first card you
pick out of the 52 cards is the
• K of hearts. You put this card aside and pick the second card from the
51 cards remaining in the deck. It is the three of diamonds. You put
this card aside and pick the third card from the remaining 50 cards in
the deck. The third card is the J of spades. Your picks are {K of hearts,
three of diamonds, J of spades}. Because you have picked the cards
without replacement, you cannot pick the same card twice.
41
42. Probability Distribution
• A probability distribution is a list of all of the possible outcomes of a
random variable along with their corresponding probability values.
42
43. Discrete Probability Distribution
• If we consider 1 and 2 as outcomes of rolling a six-sided die, then we
can’t have an outcome in between that (e.g. I can’t have an outcome
of 1.5).
• This is called probability mass function
43
44. Continuous Probability Distribution
• Sometimes we are concerned with the probabilities of random
variables that have continuous outcomes.
• Eg. The height of an adult picked at random from a population or the
amount of time that a taxi driver has to wait before their next job.
• When we use a probability function to describe a continuous
probability distribution we call it a probability density function
(commonly abbreviated as pdf).
44
45. Central Limit Theorem
• The central limit theorem states that if you have a population with
mean μ and standard deviation σ and take sufficiently large random
samples from the population with replacement text annotation
indicator, then the distribution of the sample means will be
approximately normally distributed.
45
50. Genetic Algorithm
Genetic algorithm is a search heuristic that is inspired by Charles
Darwin’s theory of natural evolution.
This algorithm reflects the process of natural selection where the fittest
individuals are selected for reproduction in order to produce offspring
of the next generation.
50
52. Phases of Genetic Algorithm
Initial population
Fitness function
Selection
Crossover
Mutation
52
53. Initial Population
The process begins with a set of individuals which is called a
Population. Each individual is a solution to the problem you want to
solve.
An individual is characterized by a set of parameters (variables) known
as Genes. Genes are joined into a string to form a Chromosome
(solution).
In a genetic algorithm, the set of genes of an individual is represented
using a string, in terms of an alphabet. Usually, binary values are used
(string of 1s and 0s). We say that we encode the genes in a
chromosome.
53
55. Fitness Function
The fitness function determines how fit an individual is (the ability of
an individual to compete with other individuals).
It gives a fitness score to each individual.
The probability that an individual will be selected for reproduction is
based on its fitness score.
55
56. Selection
The idea of selection phase is to select the fittest individuals and let
them pass their genes to the next generation.
Two pairs of individuals (parents) are selected based on their fitness
scores. Individuals with high fitness have more chance to be selected
for reproduction.
56
57. Crossover
Crossover is the most significant phase in a genetic algorithm. For each
pair of parents to be mated, a crossover point is chosen at random
from within the genes.
For example, consider the crossover point to be 3 as shown below.
57
58. Crossover
• Offspring are created by exchanging the genes of parents among
themselves until the crossover point is reached.
• The new offsprings A5 and A6 are added to the population.
58
59. Probability in crossover
• Choosing which chromosome to perform crossover
• Choosing the pair to perform crossover
• Choosing the part of chromosome to perform crossover
59
60. Mutation
• In certain new offspring formed, some of their genes can be
subjected to a mutation with a low random probability.
• This implies that some of the bits in the bit string can be flipped.
60
61. Probability in mutation
• Choosing which chromosome to perform mutation
• Choosing whether to perform mutation or not
• Choosing the part of chromosome to perform mutation
61
64. Probability usage in programming
64
# generate random floating point values
from random import seed
from random import random
# seed random number generator
seed(1)
# generate random numbers between 0-1
for _ in range(10):
value = random()
print(value)
65. Probability usage in programming
65
# generate random integer values
from random import seed
from random import randint
# seed random number generator
seed(1)
# generate some integers
for _ in range(10):
value = randint(0, 10)
print(value)
66. Probability usage in programming
66
# choose a random element from a list
from random import seed
from random import choice
# seed random number generator
seed(1)
# prepare a sequence
sequence = [i for i in range(20)]
print(sequence)
# make choices from the sequence
for _ in range(5):
selection = choice(sequence)
print(selection)
67. Probability usage in programming
67
# randomly shuffle a sequence
from random import seed
from random import shuffle
# seed random number generator
seed(1)
# prepare a sequence
sequence = [i for i in range(20)]
print(sequence)
# randomly shuffle the sequence
shuffle(sequence)
print(sequence)
69. More topics recommended to learn
• Queueing Theory
• Statistics
• Numerical Methods
• Discrete Mathematics
• Optimization problems in Operations Research
69