This document defines functions to calculate the expected value of the highest scoring item in a dataset given an acceptance rate. It integrates these functions to find the maximum expected value for different dataset sizes and acceptance rates. The analysis finds that for a dataset of size 100, the maximum expected value is achieved with an acceptance of the top 3-4 items, and that increasing the dataset size shifts the maximum to accepting a larger portion of top scoring items.
The document describes a method for summarizing the essential information of a document in 3 sentences or less. It begins by providing definitions for key terms used in the method such as sets, functions, and ordering relationships. It then provides an example application of the method to a specific problem instance, calculating an ordering relationship over subsets of a set based on a given valuation function.
The document presents information about submodular functions including:
1) It defines a submodular function v as a set function whose domain is the power set of a ground set N, and discusses properties of submodular functions.
2) It provides an example of a submodular function v with ground set N={1,2,3} and defines the polyhedron and base polyhedron associated with v.
3) It introduces the concept of a greedy algorithm for maximizing a submodular set function and outlines the steps of the greedy algorithm.
Using R in financial modeling provides an introduction to using R for financial applications. It discusses importing stock price data from various sources and visualizing it using basic graphs and technical indicators. It also covers topics like calculating returns, estimating distributions of returns, correlations, volatility modeling, and value at risk calculations. The document provides examples of commands and functions in R to perform these financial analytics tasks on sample stock price data.
This document provides estimates for several number theory functions without assuming the Riemann Hypothesis (RH), including bounds for ψ(x), θ(x), and the kth prime number pk. The following estimates are derived:
1) θ(x) - x < 1/36,260x for x > 0.
2) |θ(x) - x| < ηk x/lnkx for certain values of x, where ηk decreases as k increases.
3) Estimates are obtained for θ(pk), the value of θ at the kth prime number pk, showing θ(pk) is approximately k ln k + ln2 k - 1.
1. The document discusses various algebraic expressions and operations, including: expressions, products, values of expressions, addition, subtraction, division, and multiplication of algebraic expressions.
2. Examples are provided to demonstrate each concept, such as factorizing expressions, evaluating expressions for given values, combining like terms in additions and subtractions, and performing long division and multiplication of polynomials.
3. The key algebraic concepts covered are expressions, operations, and factorizing expressions into their prime factors or irreducible polynomials.
The document defines an ordering ≫ on the set A(v) based on the values of θ1(x), θ2(x),...,θ2n-4(x). For any x,y in A(v), x ≫ y if and only if there exists 1 ≤ k ≤ 2n-4 such that θl(x) = θl(y) for l = 1,...,k-1 and θk(x) < θk(y). It also defines the sets C(v) and properties of the functions v(S) and θl(x).
This document provides information on various algebraic concepts in Spanish including:
1) Algebraic expressions as sets of numbers and symbols connected by operational signs without functions beyond algebra.
2) Factoring algebraic expressions using common factors like (a+b)(a-b).
3) Finding the numeric value of an algebraic expression by substituting a given value.
4) Performing addition, subtraction, multiplication, and division of algebraic expressions through combining like terms or using distributive properties.
The document describes a method for summarizing the essential information of a document in 3 sentences or less. It begins by providing definitions for key terms used in the method such as sets, functions, and ordering relationships. It then provides an example application of the method to a specific problem instance, calculating an ordering relationship over subsets of a set based on a given valuation function.
The document presents information about submodular functions including:
1) It defines a submodular function v as a set function whose domain is the power set of a ground set N, and discusses properties of submodular functions.
2) It provides an example of a submodular function v with ground set N={1,2,3} and defines the polyhedron and base polyhedron associated with v.
3) It introduces the concept of a greedy algorithm for maximizing a submodular set function and outlines the steps of the greedy algorithm.
Using R in financial modeling provides an introduction to using R for financial applications. It discusses importing stock price data from various sources and visualizing it using basic graphs and technical indicators. It also covers topics like calculating returns, estimating distributions of returns, correlations, volatility modeling, and value at risk calculations. The document provides examples of commands and functions in R to perform these financial analytics tasks on sample stock price data.
This document provides estimates for several number theory functions without assuming the Riemann Hypothesis (RH), including bounds for ψ(x), θ(x), and the kth prime number pk. The following estimates are derived:
1) θ(x) - x < 1/36,260x for x > 0.
2) |θ(x) - x| < ηk x/lnkx for certain values of x, where ηk decreases as k increases.
3) Estimates are obtained for θ(pk), the value of θ at the kth prime number pk, showing θ(pk) is approximately k ln k + ln2 k - 1.
1. The document discusses various algebraic expressions and operations, including: expressions, products, values of expressions, addition, subtraction, division, and multiplication of algebraic expressions.
2. Examples are provided to demonstrate each concept, such as factorizing expressions, evaluating expressions for given values, combining like terms in additions and subtractions, and performing long division and multiplication of polynomials.
3. The key algebraic concepts covered are expressions, operations, and factorizing expressions into their prime factors or irreducible polynomials.
The document defines an ordering ≫ on the set A(v) based on the values of θ1(x), θ2(x),...,θ2n-4(x). For any x,y in A(v), x ≫ y if and only if there exists 1 ≤ k ≤ 2n-4 such that θl(x) = θl(y) for l = 1,...,k-1 and θk(x) < θk(y). It also defines the sets C(v) and properties of the functions v(S) and θl(x).
This document provides information on various algebraic concepts in Spanish including:
1) Algebraic expressions as sets of numbers and symbols connected by operational signs without functions beyond algebra.
2) Factoring algebraic expressions using common factors like (a+b)(a-b).
3) Finding the numeric value of an algebraic expression by substituting a given value.
4) Performing addition, subtraction, multiplication, and division of algebraic expressions through combining like terms or using distributive properties.
This document contains 6 problems related to Monte Carlo simulations and Brownian motion:
1. Simulates Brownian motion paths starting at 1 and 2, finds frequency of ending within 1 of each other compared to expected.
2. Uses Monte Carlo to estimate area between two functions, plots sample and compares result to exact value.
3. Examines impact of importance sampling on variance reduction. Compares standard and importance sampling methods.
4. Generates random numbers with different acceptance probabilities, plots experimental vs expected results.
5. Generates random values from two probability density functions, plots and compares acceptance rates.
6. All problems involve simulation, plotting results, and comparing to theoretical values.
1. The document provides examples of various functions in R including string functions, mathematical functions, statistical probability functions and other statistical functions. Examples are given for functions like substr, grep, sub, paste etc. to manipulate strings and functions like mean, sd, median etc. for statistical calculations.
2. Examples are shown for commonly used probability distribution functions like dnorm, pnorm, qnorm, rnorm etc. Other examples include functions for binomial, Poisson and uniform distributions.
3. The document also lists various other useful statistical functions like range, sum, diff, min, max etc. with examples. Examples are provided to illustrate the use of these functions through loops and to create a matrix.
This document provides information about quadratic functions including:
1) The general form of a quadratic function is f(x) = ax2 + bx + c, where a, b, c are constants and a ≠ 0.
2) Characteristics of quadratic functions include involving one variable only and the highest power of the variable being 2.
3) Examples are provided to demonstrate recognizing quadratic and non-quadratic functions based on their form.
1. The document discusses relations and functions, including identifying their domain and range. It provides examples of plotting points on a Cartesian plane and using graphs to represent equations.
2. Functions are defined as relations where each input is mapped to only one output. Several examples are given to demonstrate determining if a relation qualifies as a function using techniques like the vertical line test.
3. The key concepts of domain, range, and using graphs are illustrated through multiple examples of sketching and analyzing relations and functions.
The map method iterates through each element of an array and returns a new array with the results of calling a provided callback function on each element. The callback function is used to transform each element and return a new element, which gets placed in the same index of the new array. Map allows transforming each element of an array easily without using for loops. Other ways to transform arrays include forEach, for..of loops, and regular for loops but map provides a cleaner syntax for one-to-one transformations of each element.
The document discusses interpolation, which involves using a function to approximate values between known data points. It provides examples of Lagrange interpolation, which finds a polynomial passing through all data points, and Newton's interpolation, which uses divided differences to determine coefficients for approximating between points. The examples demonstrate constructing Lagrange and Newton interpolation polynomials using given data sets.
The document contains examples of factorizing polynomials and rational expressions. Various techniques are demonstrated, such as finding the highest common factor, grouping like terms, and using the difference of two squares formula.
Slides from GR8 Conf EU 2019 talk - "Groovy Refactoring Patterns". In this talk, I share the refactoring patterns I observed during Groovy development.
The document describes a Haskell program that translates characters in one string to characters in another string. It defines a translate function that maps characters from the first string (set1) to the corresponding characters in the second string (set2). A translateString function applies the translate function to a given string, and the main function gets the set1 and set2 strings from arguments, reads stdin, applies translateString, and writes the result to stdout, catching any errors.
1. The document discusses vector optimization problems and presents definitions and concepts related to nondominated solutions.
2. It introduces the concept of θ-ordering between solutions and defines what it means for one solution to be better than another based on their θ-ordering.
3. Formulas and properties are presented for calculating the θ-value of solutions based on the objective function values.
The Ring programming language version 1.10 book - Part 33 of 212Mahmoud Samir Fayed
This document provides documentation on file handling functions in the Ring programming language. It describes functions for reading and writing files, such as Read(), Write(), Dir(), Rename(), Remove(), fopen(), fclose(), as well as functions for file positioning and input/output such as fseek(), ftell(), rewind(). Examples are provided to demonstrate usage of these functions for reading file contents, writing strings to files, getting directory listings, and copying files. Mathematical functions are also briefly covered, along with functions for random number generation and working with unsigned numbers.
Clustering and Factorization using Apache SystemML by Alexandre V EvfimievskiArvind Surve
This document discusses clustering and factorization techniques in SystemML. It begins by describing k-means clustering, including how it takes as input a matrix of records and clusters them to minimize within-cluster sum of squares. It also discusses k-means++ initialization and the standard k-means algorithm. The document then describes weighted non-negative matrix factorization, which approximates a data matrix as the product of two non-negative matrices to find latent topics. It discusses optimizations for WNMF like operator fusion to reduce computation.
The document discusses drawing 2D primitives such as lines, circles, and polygons in a raster graphics system. It covers:
- Representations of lines, circles, and polygons using implicit, explicit, and parametric formulas
- Scan conversion algorithms to draw these primitives by mapping them to pixels, including basic and midpoint line algorithms, a circle midpoint algorithm, and flood fill and scan conversion approaches for polygon fill
- Components of an interactive graphics system including the application model, program, and graphics system that interfaces with display hardware like CRT and FED displays
some important questions for practice clas 12 nitishguptamaps
This document provides 15 important questions on matrices and determinants and 8 questions on continuity and differentiability for a mathematics class XII exam in 2015. The matrices and determinants section includes 1-mark, 4-mark and 13-mark questions testing various concepts like matrix multiplication, inverse of matrices, properties of determinants, and solving systems of equations using matrices. The continuity and differentiability section includes questions to test understanding of the definitions and tests for continuity and differentiability as well as solving related problems using principles of differentiation.
This document contains 6 problems related to Monte Carlo simulations and Brownian motion:
1. Simulates Brownian motion paths starting at 1 and 2, finds frequency of ending within 1 of each other compared to expected.
2. Uses Monte Carlo to estimate area between two functions, plots sample and compares result to exact value.
3. Examines impact of importance sampling on variance reduction. Compares standard and importance sampling methods.
4. Generates random numbers with different acceptance probabilities, plots experimental vs expected results.
5. Generates random values from two probability density functions, plots and compares acceptance rates.
6. All problems involve simulation, plotting results, and comparing to theoretical values.
1. The document provides examples of various functions in R including string functions, mathematical functions, statistical probability functions and other statistical functions. Examples are given for functions like substr, grep, sub, paste etc. to manipulate strings and functions like mean, sd, median etc. for statistical calculations.
2. Examples are shown for commonly used probability distribution functions like dnorm, pnorm, qnorm, rnorm etc. Other examples include functions for binomial, Poisson and uniform distributions.
3. The document also lists various other useful statistical functions like range, sum, diff, min, max etc. with examples. Examples are provided to illustrate the use of these functions through loops and to create a matrix.
This document provides information about quadratic functions including:
1) The general form of a quadratic function is f(x) = ax2 + bx + c, where a, b, c are constants and a ≠ 0.
2) Characteristics of quadratic functions include involving one variable only and the highest power of the variable being 2.
3) Examples are provided to demonstrate recognizing quadratic and non-quadratic functions based on their form.
1. The document discusses relations and functions, including identifying their domain and range. It provides examples of plotting points on a Cartesian plane and using graphs to represent equations.
2. Functions are defined as relations where each input is mapped to only one output. Several examples are given to demonstrate determining if a relation qualifies as a function using techniques like the vertical line test.
3. The key concepts of domain, range, and using graphs are illustrated through multiple examples of sketching and analyzing relations and functions.
The map method iterates through each element of an array and returns a new array with the results of calling a provided callback function on each element. The callback function is used to transform each element and return a new element, which gets placed in the same index of the new array. Map allows transforming each element of an array easily without using for loops. Other ways to transform arrays include forEach, for..of loops, and regular for loops but map provides a cleaner syntax for one-to-one transformations of each element.
The document discusses interpolation, which involves using a function to approximate values between known data points. It provides examples of Lagrange interpolation, which finds a polynomial passing through all data points, and Newton's interpolation, which uses divided differences to determine coefficients for approximating between points. The examples demonstrate constructing Lagrange and Newton interpolation polynomials using given data sets.
The document contains examples of factorizing polynomials and rational expressions. Various techniques are demonstrated, such as finding the highest common factor, grouping like terms, and using the difference of two squares formula.
Slides from GR8 Conf EU 2019 talk - "Groovy Refactoring Patterns". In this talk, I share the refactoring patterns I observed during Groovy development.
The document describes a Haskell program that translates characters in one string to characters in another string. It defines a translate function that maps characters from the first string (set1) to the corresponding characters in the second string (set2). A translateString function applies the translate function to a given string, and the main function gets the set1 and set2 strings from arguments, reads stdin, applies translateString, and writes the result to stdout, catching any errors.
1. The document discusses vector optimization problems and presents definitions and concepts related to nondominated solutions.
2. It introduces the concept of θ-ordering between solutions and defines what it means for one solution to be better than another based on their θ-ordering.
3. Formulas and properties are presented for calculating the θ-value of solutions based on the objective function values.
The Ring programming language version 1.10 book - Part 33 of 212Mahmoud Samir Fayed
This document provides documentation on file handling functions in the Ring programming language. It describes functions for reading and writing files, such as Read(), Write(), Dir(), Rename(), Remove(), fopen(), fclose(), as well as functions for file positioning and input/output such as fseek(), ftell(), rewind(). Examples are provided to demonstrate usage of these functions for reading file contents, writing strings to files, getting directory listings, and copying files. Mathematical functions are also briefly covered, along with functions for random number generation and working with unsigned numbers.
Clustering and Factorization using Apache SystemML by Alexandre V EvfimievskiArvind Surve
This document discusses clustering and factorization techniques in SystemML. It begins by describing k-means clustering, including how it takes as input a matrix of records and clusters them to minimize within-cluster sum of squares. It also discusses k-means++ initialization and the standard k-means algorithm. The document then describes weighted non-negative matrix factorization, which approximates a data matrix as the product of two non-negative matrices to find latent topics. It discusses optimizations for WNMF like operator fusion to reduce computation.
The document discusses drawing 2D primitives such as lines, circles, and polygons in a raster graphics system. It covers:
- Representations of lines, circles, and polygons using implicit, explicit, and parametric formulas
- Scan conversion algorithms to draw these primitives by mapping them to pixels, including basic and midpoint line algorithms, a circle midpoint algorithm, and flood fill and scan conversion approaches for polygon fill
- Components of an interactive graphics system including the application model, program, and graphics system that interfaces with display hardware like CRT and FED displays
some important questions for practice clas 12 nitishguptamaps
This document provides 15 important questions on matrices and determinants and 8 questions on continuity and differentiability for a mathematics class XII exam in 2015. The matrices and determinants section includes 1-mark, 4-mark and 13-mark questions testing various concepts like matrix multiplication, inverse of matrices, properties of determinants, and solving systems of equations using matrices. The continuity and differentiability section includes questions to test understanding of the definitions and tests for continuity and differentiability as well as solving related problems using principles of differentiation.
Discovering Digital Process Twins for What-if Analysis: a Process Mining Appr...Marlon Dumas
This webinar discusses the limitations of traditional approaches for business process simulation based on had-crafted model with restrictive assumptions. It shows how process mining techniques can be assembled together to discover high-fidelity digital twins of end-to-end processes from event data.
Discover the cutting-edge telemetry solution implemented for Alan Wake 2 by Remedy Entertainment in collaboration with AWS. This comprehensive presentation dives into our objectives, detailing how we utilized advanced analytics to drive gameplay improvements and player engagement.
Key highlights include:
Primary Goals: Implementing gameplay and technical telemetry to capture detailed player behavior and game performance data, fostering data-driven decision-making.
Tech Stack: Leveraging AWS services such as EKS for hosting, WAF for security, Karpenter for instance optimization, S3 for data storage, and OpenTelemetry Collector for data collection. EventBridge and Lambda were used for data compression, while Glue ETL and Athena facilitated data transformation and preparation.
Data Utilization: Transforming raw data into actionable insights with technologies like Glue ETL (PySpark scripts), Glue Crawler, and Athena, culminating in detailed visualizations with Tableau.
Achievements: Successfully managing 700 million to 1 billion events per month at a cost-effective rate, with significant savings compared to commercial solutions. This approach has enabled simplified scaling and substantial improvements in game design, reducing player churn through targeted adjustments.
Community Engagement: Enhanced ability to engage with player communities by leveraging precise data insights, despite having a small community management team.
This presentation is an invaluable resource for professionals in game development, data analytics, and cloud computing, offering insights into how telemetry and analytics can revolutionize player experience and game performance optimization.
Build applications with generative AI on Google CloudMárton Kodok
We will explore Vertex AI - Model Garden powered experiences, we are going to learn more about the integration of these generative AI APIs. We are going to see in action what the Gemini family of generative models are for developers to build and deploy AI-driven applications. Vertex AI includes a suite of foundation models, these are referred to as the PaLM and Gemini family of generative ai models, and they come in different versions. We are going to cover how to use via API to: - execute prompts in text and chat - cover multimodal use cases with image prompts. - finetune and distill to improve knowledge domains - run function calls with foundation models to optimize them for specific tasks. At the end of the session, developers will understand how to innovate with generative AI and develop apps using the generative ai industry trends.
Build applications with generative AI on Google Cloud
Secretary_Game_With_Rejection.pdf
1. In[31]:= A[k_, m_] = Gamma[m + 2] / Gamma[k + 1] / Gamma[m - k + 1];
(*expected value of x*)
xexp[k_, m_, n_] = A[k, m] / 2 *
NIntegrate[x^(m - k) * (1 - x)^k * (x * x^(1 - m + n) + (1 + x) * (1 - x^(1 - m + n))), {x, 0, 1}];
N[xexp[3, 37, 0, 100]]
NIntegrate: The integrand (1 - x)k
x-k+m
x2-m+n
+ (1 + x) 1 - x1+Times[2]+n
has evaluated to non-numerical
values for all sampling points in the region with boundaries {{0, 1}}.
NIntegrate: The integrand (1 - x)k
x-k+m
x2-m+n
+ (1 + x) 1 - x1+Times[2]+n
has evaluated to non-numerical
values for all sampling points in the region with boundaries {{0, 1}}.
Out[33]= 0.940033
In[27]:= Integrate[(n - m + 1) * y^(n - m), {y, 0, x}]
Out[27]= x1-m+n
if Re[m - n] < 1
(*now we need to adjust for the rate of being rejected*)
success[n_, m_, x_] = (1 - x^(1 - m + n));
failure[n_, m_, x_] = x^(1 - m + n);
(*r is the acceptance rate*)
(*the question we are asking
is: given that the greatest point in the test set is greater than x,
how many points are greater than x*)
(*we have to untangle success again*)
successold[n_, m_, x_] = Integrate[(n - m + 1) * y^(n - m), {y, x, 1}];
(*now given y, we can set up the space between
x and y and compare it to the space below y itself*)
(*deprecated, see below*)
(*greaterx[n_,m_,x_]=Integrate[(n-m+1)*y^(n-m)*(y-x)/y*(n-m),{y,x,1}]*)
(*for each greater than x point y,
we find the number of points that are between x and y, then add them together*)
rejected[n_, m_, x_, r_] = success[n, m, x] * (1 - r)^(greaterx[n, m, x]);
notrejected[n_, m_, x_, r_] = success[n, m, x] * (1 - (1 - r)^(greaterx[n, m, x]));
xexp[k_, m_, n_, r_] =
A[k, m] / 2 * NIntegrate[x^(m - k) * (1 - x)^k * (x * (failure[n, m, x] + rejected[n, m, x, r]) +
(1 + x) * notrejected[n, m, x, r]), {x, 0, 1}];
Plot[N[success[100, 37, x]], {x, 0, 1}, PlotRange → {-0.1, 1}]
(*makes sense, if the first number is large, it's hard to beat it*)
Plot[N[rejected[100, 37, x, 0.1]], {x, 0, 1}, PlotRange → {-0.1, 1}]
Plot[N[notrejected[100, 37, x, 0.1]], {x, 0, 1}, PlotRange → {-0.1, 1}]
5. In[140]:= Plot[xexp[k, 37, 100, 0.1], {k, 1, 15}]
Out[140]=
2 4 6 8 10 12 14
0.60
0.65
0.70
0.75
(*looks like 0.76 and only top 11 or so*)
In[141]:= Plot3D[xexp[k, m, 100, 0.1], {k, 1, 20}, {m, 1, 40}]
Out[141]=
(*it looks like it's telling me top 1 and fire after 4 trials*)
In[176]:= Quiet[FindMaximum[xexp[k, m, 100, 0.1], {k, 10}, {m, 37}]]
Out[176]=
{0.770144, {k → 11.2483, m → 35.5792}}
In[177]:= Quiet[FindMaximum[xexp[k, m, 100, 0.1], {k, 1}, {m, 4}]]
Out[177]=
{0.794735, {k → 3.64866, m → 14.252}}
In[179]:= Quiet[FindMaximum[{xexp[k, m, 100, 0.1], 1 ≤ k ≤ 15, 1 ≤ m ≤ 40}, {k, m}]]
Out[179]=
{0.794735, {k → 3.64867, m → 14.252}}
5
6. (*seems like there could be other local maxima but the global one is at k=
3.65 and m = 14.3*)
In[180]:= Quiet[FindMaximum[{xexp[k, m, 1000, 0.1], 1 ≤ k ≤ 50, 1 ≤ m ≤ 400}, {k, m}]]
Out[180]=
{0.962388, {k → 9.65589, m → 171.697}}
In[184]:= Quiet[FindMaximum[{xexp[k, m, 10 000, 0.1], 10 ≤ k ≤ 50, 1000 ≤ m ≤ 2000}, {k, m}]]
Out[184]=
{0., {k → 19.2022, m → 1085.2}}
(*and then it just broke*)
In[185]:= Quiet[FindMaximum[{xexp[k, m, 200, 0.1], 1 ≤ k ≤ 15, 1 ≤ m ≤ 40}, {k, m}]]
Out[185]=
{0.871543, {k → 5.4209, m → 31.5675}}
In[188]:= Quiet[FindMaximum[{xexp[k, m, 300, 0.1], 1 ≤ k ≤ 15, 1 ≤ m ≤ 60}, {k, m}]]
Out[188]=
{0.904275, {k → 6.48495, m → 48.9963}}
(*another thing to consider is that (1-r)^(greaterx[n,m,x]) is log-normal,
not normal*)
In[191]:= greaterx["mean"][n_, m_, x_] = Integrate[(n - m + 1) * y^(n - m) * (y - x) / y * (n - m), {y, x, 1}]
greaterx["var"][n_, m_, x_] = Integrate[(n - m + 1) * y^(n - m) * (y - x) / y * x / y * (n - m), {y, x, 1}]
Out[191]=
-
(-m + n) (1 - m + n) m + n (-1 + x) + x - m x - x1-m+n
(-1 + m - n) (m - n)
if condition
Out[192]=
-
(-m + n) (1 - m + n) x 1 + m + n (-1 + x) - m x - x-m+n
(m - n) (1 + m - n)
if condition
In[197]:= (*I need the mean of (1-r)^(Gaussian) - it's mu + var/2. aside: we have ln[term]=
log(1-r)*Gaussian, this factor doesn't influence mean or std*)
greaterx[n_, m_, x_] = greaterx["mean"][n, m, x] + greaterx["var"][n, m, x] / 2
Out[197]=
-
(-m + n) (1 - m + n) x 1 + m + n (-1 + x) - m x - x-m+n
2 (m - n) (1 + m - n)
-
(-m + n) (1 - m + n) m + n (-1 + x) + x - m x - x1-m+n
(-1 + m - n) (m - n)
if condition
6
14. (*so the median person has it considerably worse than the mean person -
which one am I*)
(*I think I am the mean person. The reason is that the log-
normal distribution gets integrated over all outcomes of the first trial-
phase so that means it becomes Gaussian again
due to the Central Limit Theorem - am I wrong?*)
14