This document provides an overview of probability distributions and related concepts. It defines key probability distributions like the binomial, beta, multinomial, and Dirichlet distributions. It also describes probability distribution equations like the cumulative distribution function and probability density function. Additionally, it outlines descriptive parameters for distributions like mean, variance, skewness and kurtosis. Finally, it briefly discusses probability theorems such as the law of large numbers, central limit theorem, and Bayes' theorem.
A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution. Suppose we flip a coin two times and count the number of heads (successes).
According to Wikipedia point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population means).
A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution. Suppose we flip a coin two times and count the number of heads (successes).
According to Wikipedia point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population means).
Probability
Random variables and Probability Distributions
The Normal Probability Distributions and Related Distributions
Sampling Distributions for Samples from a Normal Population
Classical Statistical Inferences
Properties of Estimators
Testing of Hypotheses
Relationship between Confidence Interval Procedures and Tests of Hypotheses.
Chapter 4 part3- Means and Variances of Random Variablesnszakir
Statistics, study of probability, The Mean of a Random Variable, The Variance of a Random Variable, Rules for Means and Variances, The Law of Large Numbers,
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UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
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In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
2. Topics
Probability Distribution
Probability Distribution Equations
Descriptive parameters for Probability Distributions
Probability Theorems
Binary Variables
The beta distribution
Multinomial Variables
The Dirichlet distribution
3. Probability Distribution
A function that describes all the possible values and
likelihoods that a random variable can take within a
given range.
This range will be between the minimum and
maximum possible values, but where the possible
value is likely to be plotted on the probability
distribution depends on a number of factors, including
the distributions mean, standard deviation, skewness
and kurtosis.
4. Probability Distributions
Equations
The section on probability equations explains the
equations that define probability distributions.
Cumulative distribution function (cdf)
Probability mass function (pmf)
Probability density function (pdf)
5. Probability Distributions
Equations
Cumulative distribution function (cdf)
The (cumulative) distribution function, or probability
distribution function, F(x) is the mathematical equation
that describes the probability that a variable X is less
that or equal to x, i.e.
F(x) = P(X≤x) for all x
where P(X≤x) means the probability of the event X≤x.
7. Probability Distributions
Equations
A cumulative distribution function has the following
properties:
F(x) is always non-decreasing, i.e.
F(x) = 0 at x = -∞ or minimum
F(x) = 1 at x = ∞ or maximum
( ) 0
d
F x
dx
8. Probability Distributions
Equations
Probability mass function (pmf)
If a random variable X is discrete, i.e. it may take any of
a specific set of n values xi, i = 1 to n, then:
P(X=xi) = p(xi)
p(x) is called the probability mass function
9. Probability Distributions
Equations
The graph of a probability mass function. All the
values of this function must be non-negative and
sum up to 1.
The probability mass function of a fair die. All the
numbers on the die have an equal chance of
appearing on top when the die stops rolling.
1 3 7
0.2 0.5 0.3
1 2 3 4 5 6
1/6 1/6 1/6 1/6 1/6 1/6
11. Probability Distributions
Equations
Probability density function (pdf)
If a random variable X is continuous, i.e. it may take any
value within a defined range (or sometimes ranges),
the probability of X having any precise value within that
range is vanishingly small because a total probability of
1 must be distributed between an infinite number of
values. In other words, there is no probability mass
associated with any specific allowable value of X.
12. Probability Distributions
Equations
Instead, we define a probability density function f(x) as:
i.e. f(x) is the rate of change (the gradient) of the
cumulative distribution function. Since F(x) is always
non-decreasing, f(x) is always non-negative.
( ) ( )
d
f x F x
dx
13. Probability Distributions
Equations
For a continuous distribution we cannot define the
probability of observing any exact value. However, we
can determine the probability of lying between any two
exact values (a, b):
where b>a.
( ) ( ) ( )P a x b F b F a
14. Descriptive parameters for
Probability Distributions
The section on probability parameters explains the
meaning of standard statistics like mean and variance
within the context of probability distributions.
15. Descriptive parameters for
Probability Distributions
Location
Mode: is the x-value with the greatest
probability p(x) for a discrete distribution, or the
greatest probability density f(x) for a continuous
distribution.
Median: is the value that the variable has a 50%
probability of exceeding, i.e. F(x50) = 0.5
16. Descriptive parameters for
Probability Distributions
Mean : also known as the expected value, is
given by:
for discrete variables
for continuous variables
The mean is known as the first moment about zero. It
can be considered to be the centre of gravity of the
distribution.
1
n
i i
i
x p
. ( ).x f x dx
17. Descriptive parameters for
Probability Distributions
Spread
Standard Deviation: measures the amount of
variation or dispersion from the average or mean. The
standard deviation is the positive square root of the
variance.
The standard deviation has the same dimension as the
data, and hence is comparable with deviations of the
mean.
18. Descriptive parameters for
Probability Distributions
Variance: measures how far a set of numbers is
spread out.
An equivalent measure is the square root of the
variance, called the standard deviation.
The variance is one of several descriptors of
a probability distribution. In particular, the variance is
one of the moments of a distribution.
19. Descriptive parameters for
Probability Distributions
Shape
Skewness:
The skewness statistic is calculated from the following
formulae:
Discrete variable:
Continuous variable:
max
3
min
3
( ) . ( ).x f x dx
S
3
1
3
( ) .
n
i i
i
x p
S
20. Descriptive parameters for
Probability Distributions
Kurtosis:
The kurtosis statistic is calculated from the following
formulae:
Discrete variable:
Continuous variable:
max
4
min
4
( ) . ( ).x f x dx
K
4
1
4
( ) .
n
i i
i
x p
K
21. Probability Theorems
Probability theorems explains some fundamental
probability theorems most often used in modelling risk,
and some other mathematical concepts that help us
manipulate and explore probabilistic problems.
The strong law of large numbers
Central limit theorem
Binomial Theorem
Bayes theorem
22. Probability Theorems
The strong law of large numbers
The strong law of large numbers says that the larger
the sample size (i.e. the greater the number of
iterations), the closer their distribution (i.e. the risk
analysis output) will be to the theoretical
distribution (i.e. the exact distribution of the models
output if it could be mathematically derived).
23. Probability Theorems
Central Limit Theorem(CLT)
The distribution of the sum of N i.i.d. random
variables becomes increasingly Gaussian as N
grows.
Example: N uniform [0,1] random variables.
24. Probability Theorems
Binomial Theorem
a Formula for finding any power of a binomial without
multiplying at length.
Properties of binomial coefficient!
!( )!
n n
x x n x
0
1
1
0
n
i
n n
n x x
n n n
x x n x
n n
n
a b a b
n b n i
25. Probability Theorems
Bayes theorem
a theorem describing how the conditional probability
of each of a set of possible causes for a given observed
outcome can be computed from knowledge of the
probability of each cause and the conditional
probability of the outcome of each cause.
27. Binary Variables
Binary variable Observations (i.e., dependent variables)
that occur in one of two possible states,
often labelled zero and one. E.g., “improved/not
improved” and “completed
task/failed to complete task.”
Coin flipping: heads=1, tails=0
Bernoulli Distribution
( 1| )p x
1
( | ) (1 )
var 1
x x
Bern x
x
x
28. Binary Variables
N coin flips
Binomial distribution
( | , )p m heads N
0
2
0
( | , ) ( ) (1 )
( | , )
var[ ] ( [ ]) ( | , ) (1 )
m N m
m
N
m
N
m
Bin m N N
m mBin m N N
m m m Bin m N N
29. Beta distribution
Beta is a continuous distribution defined on the interval
of 0 and 1, i.e.,
parameterized by two positive parameters a and b.
where T(*) is gamma function. beta is conjugate to the
binomial and Bernoulli distributions
0,1
11
2
| , 1
var
1
ba
Beta
a b
a b
a b
a
a b
ab
a b a b
30. Beta distribution
Illustration of one step of sequential Bayesian
inference. The prior is given by a beta distribution
with parameters a = 2, b = 2, and the likelihood function,
given by (2.9) with N = m = 1, corresponds to a
single observation of x = 1, so that the posterior is given by
a beta distribution with parameters a = 3, b = 2.
32. Multinomial Distribution
Multinomial distribution is a generalization of the
binominal distribution. Different from the binominal
distribution, where the RV assumes two outcomes, the RV
for multi-nominal distribution can assume k (k>2) possible
outcomes.
Let N be the total number of independent trials, mi,
i=1,2, ..k, be the number of times outcome i appears.
Then, performing N independent trials, the probability
that outcome 1 appears m1, outcome 2, appears m2,
…,outcome k appears mk times is
33. Multinomial Distribution
1 2
11 2
, ,...., | ,
...
var 1
cov
K
mK
K K
KK
K K
K K
j K j K
Mult
N
m m m N
m m m
m N
m N
m m N
34. The Dirichlet Distribution
The Dirichlet distribution is a continuous multivariate
probability distributions parametrized by a vector of
positive reals a. It is the multivariate generalization of the
beta distribution.
Conjugate prior for the
multinomial distribution.
10
11
1
( | )
...
0
K
K
k
kK
K
k
k
Dir
Editor's Notes
One role for the distributions discussed in this chapter is to model the probability
distribution p(x) of a random variable x, given a finite set x1, . . . , xN of
observations.
This problem is known as density estimation.
For the purposes of
this chapter, we shall assume that the data points are independent and identically
distributed.
the cumulative distribution function (CDF), or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found to have a value less than or equal to x.
In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.