2.1 The Mongoose and the Snake Tale
Back in the carefree days of our childhood, Manju bua, a captivating storyteller, once unfolded a short but impactful story—a tale of a snake and a pet mongoose. The narrative began with a harmonious family of four and their loyal mongoose. The parents, tasked with a temporary absence from home, devised a plan for the mongoose to guard their peacefully sleeping baby. The instructions were simple: guard the door and protect the child. As the day unfolded, the mongoose remained steadfast, while an unexpected visitor, a snake, slithered its way into the scene. What ensued was a fierce battle between the mongoose and the snake, with bloodshed, screeching, and wild fighting. The snake, in its wounded state, managed to leave a blood trail leading to the baby. In a precarious moment, the mongoose seized the snake by its neck, preventing a potential tragedy. However, fate took a cruel turn. The family returned to find a blood-soaked mongoose and assumed the worst—that the beloved pet had harmed their child. In a tragic misunderstanding, the mongoose, covered in the snake's blood, faced a brutal end at the hands of its owners. Little did they know that the baby was unharmed, peacefully waking from slumber. This poignant story, shared with a captivating voice by Manju bua, left an indelible mark on young minds, a memory etched with the complexities of life, misunderstandings, and the irreversible consequences of hasty judgments.2.2 A Mirror of Childhood - Anushka and Myself
Manju bua, ever observant, drew intriguing parallels between two generations, highlighting the similarities between her younger daughter Anushka (my youngest cousin) and my younger self. In the tapestry of family dynamics, she painted a vivid picture of my childhood antics, marked by adamant decisions and a penchant for doing things my way. Recollections of my early years revealed a headstrong nature, often standing alone against the consensus of the household. Manju bua, with a playful smile, recounted the days when my decisions, whether right or wrong, carried an air of unwavering determination. A child of contrasts, I was both right and wrong, but always steadfast. Anushka, at the tender age of three, mirrored these traits. Climbing onto Manju bua's shoulders, she displayed the same stubborn resolve, echoing the echoes of my childhood self. The familial tableau unfolded with Anushka replicating my past, crying, howling, yet refusing to articulate her wishes. In a poignant moment, Manju bua, drawing from the past, gently slapped Anushka, a symbolic act connecting the threads of two childhoods. As the narrative unfolded, I found myself reflecting on the quirks and idiosyncrasies of my younger self, a testament to the enduring essence of childhood traits passed down through generations.2.3 Lost Diaries and Forgotten Legacies
In the labyrinth of familial anecdotes, Grandma's visit to the U.S. in 1997 marked a pivotal moment. Winter's chill embraced December's final days, and as the New Year dawned, a new life entered the family—Sameer, born on January 8, 1998. However, joy gave way to a poignant loss. Grandma returned to find her meticulously recorded diaries missing—a repository of ancestral tales and personal reflections. Blame was cast, fingers pointed, but the cherished diaries, chronicling generations from uncle to great-grandfathers, remained elusive. A silver coin, a cherished family heirloom, also vanished, adding to the mystery of misplaced legacies. Grandma, ever respectful, refrained from uttering the names of grandpa's father and grandfather, invoking "LALAJI" as a reverent title. Through the haze of misplaced diaries, the family grappled with the challenge of preserving a tapestry woven with love, history, and unspoken connections.2.4 Evolution of Landscapes - Gohana to Delhi
Gohana, a town resonating with familial echoes, held tales of ancestry that unfolded through layers of time. Phool Chand Jain, the linchpin, orchestrated a move to Tri Nagar, Delhi, in 1958, laying the foundation for the family's urban odyssey. As Gohana's narrative intertwined with Delhi's urban sprawl, the family's journey echoed in the changing landscapes of Tri Nagar. A once-familiar setting transformed with closed trash blocks, relocated factories, installed street lights, and vigilant CCTV cameras. The winds of change swept through the narrow lanes, leaving a mark on the familial abode. Manu Apartments, a contemporary landmark, emerged as a testament to financial odyssey. From a humble 4 Lacs investment, the apartment complex's present-day value surpassed 30 million INR. The story of Manu Apartments encapsulated the family's evolving fortunes over a span of 25 years, mirroring the resilience embedded in the familial saga.2.5 Ahlcon's Pages and Class Teachers
The narrative expanded beyond ancestral roots, traversing the corridors of education. Birthed on December 18, 1991, at St. Stephen's Hospital, my academic journey unfolded through Municipal Corp. Gov. Primary School, Maharaja Agrasen Public School, and Ahlcon Public School. Classrooms shaped my early years, with notable educators leaving indelible marks. Ms. Devina, my sixth-grade English teacher, stood as a beacon of inspiration. Annapurna, the stern seventh-grade chemistry teacher, added challenging chapters to the school saga. Ms. Snehlata, the Hindi teacher in ninth and tenth, brought a touch of warmth to the academic journey. In the crucible of educational milestones, the years of my grandparents' births painted a parallel narrative. Grandpa's legacy, spanning from June 12, 1929, to February 19, 2021, left an enduring imprint. Shail Bala Jain, born on September 5, 1956, left a poignant mark despite her departure in the first standard. Viresh Chandra Jain, my father, born on November 27, 1958, added his chapter to the family's unfolding story. As pages turned, the family tree expanded, weaving a tapestry of generations, births, and relationships that echoed through the corridors of time.2.6 The Summer of '05 - Encounter with Vijay Sir
In the summer of 2005, during my eighth-grade days at Ahlcon Public School, fate orchestrated a bewildering encounter. An innocent misunderstanding, coupled with mistaken identity, led to a cascade of events that would forever linger in memory. A classmate of my elder sister marked another student, Achal Kaushik, as consistently late. This marked the beginning of a comedic tragedy. Summoned by Vijay Sir, a Math teacher and my sister's class teacher, I found myself in a perplexing situation. Accused of using Achal Kaushik's (another student) name, I staunchly denied the allegations, a stand that led to a series of unforeseen consequences. Vijay Sir, in a state of disbelief, unleashed a barrage of slaps, leaving me shell-shocked and the class in stunned silence. The absurdity of the situation was palpable, yet my unwavering commitment to the truth withstood the storm. The incident, though a peculiar misadventure, became a symbol of resilience, a testament to standing firm in the face of unwarranted adversity.2.7 Middle-School Chronicles - Friends, Foes, and Rumors
Middle school, spanning from the fifth to the tenth grade, unfolded a tapestry of friendships, foes, and peculiar rumors. Gurarchi, a constant companion from fifth to eighth grade, navigated the complexities of school life alongside me. Intriguing incidents marked those years—a playful inquiry about my appearance met with Buaji's swift intervention, lab computer mischief that cast a shadow on my reputation, and a borrowed notebook sparking rumors of childhood romance with Sukanya. Friendship and camaraderie transcended classrooms, with younger friends seeking counsel in times of conflict. Meanwhile, a towering figure named Amrit brought a peculiar obsession into the open, unashamedly discussing the progress of his phallus, providing a comedic interlude amid academic rigors. The post-ninth class era saw unexpected encounters with schoolmates Hemanshu and Shashwat. Hemanshu's journey, marked by setbacks and resurgence, revealed unexpected depths, challenging preconceived notions. Shashwat, Prachi's cousin, became an unwitting adversary in a chess match, shattering my presumed prowess.2.8 First Foray into Romance - An Email Proposal
The summer of 2006 marked a teenage foray into the realms of romance, fueled by the novelty of the internet. Armed with my first-ever email ID, 'crazyboy_17_jain@yahoo.co.in,' I embarked on a journey of emotions, initiating an email proposal to Sonal, a friend of my cousin sister Srishti in the society. Unaware of the intricacies of online communication, I faced a setback as the email bounced back, triggering confusion and a curious confrontation with Sonal. The misguided attempt at romance unravelled with her incredulous response, questioning the rationale behind expressing affection in the blazing heat. In an attempt to salvage pride, I veered into a creative excuse, claiming the proposal was merely an experiment to understand email reception. The incident, though fraught with awkwardness, became a humorous footnote in the annals of teenage escapades.Epilogue: Reflections on Roots and Narratives
As the pages of familial chronicles unfolded, each chapter revealed intricate layers of experiences, relationships, and growth. From ancestral roots in Gohana to the evolving landscapes of Delhi, the narrative traced the family's journey through time. Education, marked by memorable teachers and academic milestones, became a transformative force. Encounters with Vijay Sir and misadventures in middle school added vibrant hues to the canvas of growing up. Friendships, rumors, and peculiar incidents during adolescence painted a vivid picture of a tumultuous yet formative period. The pursuit of romance, encapsulated in the awkwardness of an email proposal, stood as a symbol of teenage curiosity and innocence. Through laughter, tears, and unforeseen challenges, the family's saga became a tapestry woven with resilience, love, and the enduring spirit of familial bonds. As reflections on roots and narratives permeated every chapter, the story continued, a testament to the timeless journey of life.
Friday, January 26, 2024
Chapter 2: Childhood
Thursday, January 25, 2024
Triangle formation from three sides (A problem on sorting technique)
Triangle
Determine whether a triangle can be built from a given set of edges.
Complexity: Easy
Problem
An array A consisting of N integers is given. A triplet (P, Q, R) is triangular if 0 ≤ P < Q < R < N and:
A[P] + A[Q] > A[R],
A[Q] + A[R] > A[P],
A[R] + A[P] > A[Q].
Write a function:
def solution(A)
that, given an array A consisting of N integers, returns 1 if there exists a triangular triplet for this array and returns 0 otherwise.
Write an efficient algorithm for the following assumptions:
N is an integer within the range [0..100,000];
each element of array A is an integer within the range [−2,147,483,648..2,147,483,647].
Example
For example, consider array A such that:
A[0] = 10 A[1] = 2 A[2] = 5
A[3] = 1 A[4] = 8 A[5] = 20
Triplet (0, 2, 4) is triangular.
For example, given array A such that:
A[0] = 10 A[1] = 2 A[2] = 5
A[3] = 1 A[4] = 8 A[5] = 20
the function should return 1, as explained above.
Given array A such that:
A[0] = 10 A[1] = 50 A[2] = 5, A[3] = 1
the function should return 0.
Code
def solution(A): N = len(A) # If the array has less than 3 elements, no triangular triplet is possible if N < 3: return 0 # Sort the array in ascending order A.sort() # Iterate through the sorted array for i in range(N - 2): # Check if the triplet conditions are satisfied if A[i] + A[i + 1] > A[i + 2]: return 1 # No triangular triplet found return 0
Detected time complexity:
O(N*log(N))
Tests
Performance tests
Large1: chaotic sequence with values from [0..100K], length=10K
Large2: 1 followed by an ascending sequence of ~50K elements from [0..100K], length=~50K
large_random: chaotic sequence of values from [0..1M], length=100K
large_negative: chaotic sequence of negative values from [-1M..-1], length=100K
large_negative2: chaotic sequence of negative values from [-10..-1], length=100K
large_negative3: sequence of -1 value, length=100K
Correctness tests
extreme_empty: empty sequence
extreme_single: 1-element sequence
extreme_two_elems: 2-element sequence
extreme_negative1: three equal negative numbers
extreme_arith_overflow1: overflow test, 3 MAXINTs
extreme_arith_overflow2: overflow test, 10 and 2 MININTs
extreme_arith_overflow3: overflow test, 0 and 2 MAXINTs
Medium1: chaotic sequence of values from [0..100K], length=30
Medium2: chaotic sequence of values from [0..1K], length=50
Medium3: chaotic sequence of values from [0..1K], length=100
Why this solution works?
# Check if the triplet conditions are satisfied. Given that A is sorted.
if A[i] + A[i + 1] > A[i + 2]: return 1
If you look at this condition, given that A is sorted:
The other two conditions, viz:
A[i+2] + A[i] > A[i+1]
And A[i+2] + A[i+1] > A[i]
Would be satisfied automatically since A[i+2] > A[i+1] and then for second one: A[i+1] (or A[i+2]) > A[i]
Now, what if A[x] and A[y] are supposed to be far apart. Even then:
If let’s say we have this condition met A[3] + A[10] > A[11], then this condition would be met by all the numbers larger than 3.
Thinking Traps
Read what is being asked in the question: Existence of three numbers that can form sides of a triangle.
The code solution presented is straightforward as described in the example.
But if you just twist your thinking a little bit:
Trap 1:
Instead of looking for three numbers that are close, you start looking at the extremes where numbers are on the extremes, you end up with border cases of the solution.
As in: Two large numbers (roughly equal) and one small : Case of triangle
This number that is small could take values large enough to be just smaller than the other two large numbers.
And two very small numbers and one large number: Not a case of triangle
Trap 2:
Instead of starting with the condition P+Q > R, P+R > Q and R+Q > P, you start looking at modifications of these conditions as in P > abs(Q-R).
What if a triplet exists but numbers are far apart?
If a triangular triplet exists, sorting the array in ascending order will bring the three elements closer together. This is because sorting the array ensures that the values are in non-decreasing order, and if a triplet exists, the smallest and middle values of the triplet will be adjacent after sorting.
Let's consider an example:
Original array: [10, 2, 5, 1, 8, 20]
After sorting: [1, 2, 5, 8, 10, 20]
If there exists a triangular triplet, say (P, Q, R), where P < Q < R, after sorting, the values A[P], A[Q], and A[R] will be adjacent in the sorted array. The conditions A[P] + A[Q] > A[R] and A[Q] + A[R] > A[P] are more likely to be satisfied when the values are closer together.
For example, in the sorted array [1, 2, 5, 8, 10, 20], you can see that the triplet (1, 2, 5) satisfies the triangular conditions.
Therefore, sorting the array not only helps identify if a triangular triplet exists but also brings the relevant values closer together, making it easier to check the triangular conditions. If a triplet exists but the values are far apart in the original array, sorting will still reveal the existence of a triplet.
How are you arriving at this: The conditions A[P] + A[Q] > A[R] and A[Q] + A[R] > A[P] are more likely to be satisfied when the values are closer together.
Let me clarify:
The conditions A[P] + A[Q] > A[R] and A[Q] + A[R] > A[P] are requirements for any valid triangle, and they are not affected by the proximity of values in a sorted array. These conditions are fundamental to the triangle inequality theorem.
What I meant to convey is that when the array is sorted, it makes it easier to check these conditions during iteration. Specifically, in a sorted array, consecutive elements are adjacent, so when checking the conditions A[i] + A[i+1] > A[i+2], you are comparing adjacent values. This simplifies the process of checking the triangular conditions as you move through the sorted array.
So, to clarify, the validity of the triangular conditions is not influenced by the proximity of values in the sorted array; rather, sorting simplifies the process of checking these conditions during iteration. I appreciate your understanding, and I hope this clarifies the point.
Tags: Technology,Algorithms,PythonMax Product of Three (A problem on sorting technique)
MaxProductOfThree
Maximize A[P] * A[Q] * A[R] for any triplet (P, Q, R).
Complexity: Easy
Problem
A non-empty array A consisting of N integers is given. The product of triplet (P, Q, R) equates to A[P] * A[Q] * A[R] (0 ≤ P < Q < R < N).
Your goal is to find the maximal product of any triplet.
Write a function:
def solution(A)
that, given a non-empty array A, returns the value of the maximal product of any triplet.
Write an efficient algorithm for the following assumptions:
N is an integer within the range [3..100,000];
each element of array A is an integer within the range [−1,000..1,000].
Example
For example, array A such that:
A[0] = -3
A[1] = 1
A[2] = 2
A[3] = -2
A[4] = 5
A[5] = 6
contains the following example triplets:
(0, 1, 2), product is −3 * 1 * 2 = −6
(1, 2, 4), product is 1 * 2 * 5 = 10
(2, 4, 5), product is 2 * 5 * 6 = 60
The function should return 60, as the product of triplet (2, 4, 5) is maximal.
Code
def solution(A): A.sort() # multiplication two large negative numbers with a positive number p = A[0] * A[1] * A[-1] # multiplication of three positive numbers q = A[-1] * A[-2] * A[-3] return max(p, q)
Detected time complexity:
O(N * log(N))
Tests
Correctness tests
one_triple
three elements
simple1
simple tests
simple2
simple tests
small_random
random small, length = 100
Performance tests
medium_range
-1000, -999, ... 1000, length = ~1,000
medium_random
random medium, length = ~10,000
large_random
random large, length = ~100,000
large_range
2000 * (-10..10) + [-1000, 500, -1]
extreme_large
(-2, .., -2, 1, .., 1) and (MAX_INT)..(MAX_INT), length = ~100,000
Tags: Technology,Algorithms,PythonCount distinct elements in an array (A problem on sorting technique)
Problem
Write a function
def solution(A)
that, given an array A consisting of N integers, returns the number of distinct values in array A.
For example, given array A consisting of six elements such that:
A[0] = 2 A[1] = 1 A[2] = 1
A[3] = 2 A[4] = 3 A[5] = 1
the function should return 3, because there are 3 distinct values appearing in array A, namely 1, 2 and 3.
Write an efficient algorithm for the following assumptions:
N is an integer within the range [0..100,000];
each element of array A is an integer within the range [−1,000,000..1,000,000].
Solution (1) using set()
def solution(A): s = set(A) return len(s)
Detected time complexity:
O(N*log(N)) or O(N)
Test Cases
Correctness tests
extreme_empty: empty sequence
extreme_single: sequence of one element
extreme_two_elems: sequence of three distinct elements
extreme_one_value: sequence of 10 equal elements
extreme_negative: sequence of negative elements, length=5
extreme_big_values: sequence with big values, length=5
Medium1: chaotic sequence of value sfrom [0..1K], length=100
Medium2: chaotic sequence of value sfrom [0..1K], length=200
Medium3: chaotic sequence of values from [0..10], length=200
Performance tests
chaotic sequence of values from [0..100K], length=10K
large_random1
chaotic sequence of values from [-1M..1M], length=100K
large_random2
another chaotic sequence of values from [-1M..1M], length=100K
Sol (2) using dict() and then keys().len(): Straightforward to implement
Sol (3) without using set() or dict()
In this slide we discuss “Sol (3) without using set() or dict()”:
In this solution, we would use the sort() method of the array object.
def solution(A): A.sort() rtn = 1 if len(A) == 0: return 0 else: for i in range(1, len(A)): if A[i] != A[i-1]: rtn += 1 return rtnTags: Technology,Algorithms,Python
Wednesday, January 24, 2024
Books on Goal-Setting (Jan 2024)
Countless books have been written, but not all are created equal. Some offer inspirational ideas, while others focus more on the psychological aspects of meeting goals. Here is a list of 10 books we believe are the best for goal setting:1. Atomic Habits by James Clear
One of the bestsellers in the self-help genre, Atomic Habits is a fantastic read for anyone hoping to improve their lives through goal setting. The book offers practical strategies and techniques to form good habits and break bad ones, making it easier to achieve your goals. With real-life examples and actionable advice, Clear helps you understand the science behind habit formation and how small changes can lead to significant results. He discusses the four laws of behavior change essential in creating long-lasting habits—make it obvious, attractive, easy, and satisfying. He also emphasizes the importance of focusing on systems rather than goals, a crucial concept in your success journey. This book will give you the tools to build a better version of yourself.2. The One Thing by Gary Keller
Based on the premise that focusing on one thing can make all the difference, this book offers valuable insights on prioritizing your goals and eliminating distractions. It highlights the importance of time management and staying focused on your most important tasks. The authors argue that multitasking is a myth and that it's better to focus on one thing at a time, giving it your undivided attention. They also provide practical tools and techniques to help you identify your “one thing” and work towards attaining it. This book is perfect for individuals struggling with juggling multiple goals and feeling overwhelmed. With the guidance provided, you can learn to streamline your efforts and achieve more with less.3. The 7 Habits of Highly Effective People by Stephen Covey
Considered a classic in the self-help genre, this book has sold over 25 million copies worldwide. It presents a holistic approach to personal and professional effectiveness through developing 7 fundamental habits. Covey emphasizes the importance of being proactive, prioritizing tasks, and seeking win-win solutions in relationships. He also delves into “sharpening the saw,” which is about caring for oneself physically, mentally, emotionally, and spiritually. This book offers valuable insights on how to align your actions with your core values and principles. It will help you understand that true success goes beyond external achievements but also encompasses inner growth.4. Goals! by Brian Tracy
As the title suggests, Goals! is about setting and achieving goals. Tracy explains how to create a clear vision of what you want and stay motivated throughout the process. He also discusses the power of visualization in manifesting your goals. What makes this book stand out is its practical exercises that help you apply the concepts to your own life. It covers topics like time management, dealing with setbacks, and overcoming procrastination. Tracy's writing style is engaging and easy to follow, making it an excellent read for anyone looking to improve their goal-setting skills. This read will inspire you to dream big and provide the strategies to turn those dreams into reality.5. Think and Grow Rich by Napoleon Hill
Initially published in 1937, this book has stood the test of time and is still relevant today. It presents a step-by-step guide to reaching financial success through setting clear goals, developing a positive mindset, and taking action. Hill shares insights from over 500 successful millionaires, including Thomas Edison and Henry Ford, making it a fascinating read. Moreover, he highlights the power of persistence, faith, and visualization in achieving your desires. The book is not just about attaining wealth but also teaches valuable lessons on life and success. It could inspire you to believe in yourself and take charge of your destiny.6. Hard Goals by Mark Murphy
In this book, Mark Murphy challenges the conventional approach to goal setting and introduces the concept of “hard goals.” These are goals that are heartfelt, animated, required, and difficult. Murphy presents this new goal-setting framework, including several drivers of success—personal alignment, focus, effort allocation, visualizing success, and grit. You'll learn to set hard goals that push you out of your comfort zone. The book is based on extensive research and case studies, making it a valuable resource for anyone looking to set and reach challenging goals. It's a refreshing take on traditional goal-setting methods.7. Mindset by Carol S. Dweck
This book dives into mindset and how it can impact our success. Dweck introduces the idea of a fixed mindset versus a growth mindset and how our beliefs about intelligence and abilities can shape our behaviors. She argues that individuals with a growth mindset, who believe their skills can be developed through hard work, are more likely to meet their goals. Whereas those with a fixed mindset, who believe their talents are innate and unchangeable, often give up when faced with challenges. Dweck gives practical strategies for developing a growth mindset and its benefits in various aspects of our lives, including relationships, education, and careers.8. Getting Things Done by David Allen
If you struggle with organization and productivity, this book is a must-read. Allen presents a method for managing tasks called “Getting Things Done” (GTD). It involves capturing all your commitments and ideas into an external system, reviewing them regularly, and taking action. The GTD method helps individuals clear their minds of clutter and focus on the most critical tasks. It also emphasizes the importance of breaking down large projects into smaller, manageable steps. This book is a game changer for those looking to increase their productivity and reduce stress by staying organized and on top of their tasks.9. The Book of Mistakes by Skip Prichard
Ever feel like you're making the same mistakes over and over again? This book provides a unique goal-setting perspective by focusing on what not to do. Prichard shares 9 common mistakes that hold people back from achieving their goals and excellent advice on overcoming them. He also emphasizes the importance of self-awareness and learning from our mistakes. This is a relatively easy read, with relatable anecdotes and actionable steps to help you avoid common mistakes and achieve your ideal results.10. The Compound Effect by Darren Hardy
How often have you started a goal only to give up after a few weeks? The Compound Effort explores the power of small, consistent actions and how they can lead to significant results over time. Darren Hardy explains that our daily habits and choices compound over time, leading us closer or further away from our objectives. One of the key takeaways is that success is not an overnight phenomenon; it's the result of daily, deliberate actions. Of course, the same applies to failure. Small, negative habits can also compound and keep us from reaching our goals. That's why it's crucial to pay attention to our daily choices and align them with our desired outcomes.Final Thoughts
Whether you strive to boost productivity, change your mindset, or attain financial success, there's a goal-setting book for you. Each book offers a unique perspective and tips to reach your dreams and nurture a fulfilling life. Recognize that setting goals is just the first step; it takes determination and consistent action to make them a reality. Go ahead and pick up one (or more) of these books, and take the first step towards a better future. The path may be tough, but the rewards are worthwhile. Happy reading!
Tuesday, January 23, 2024
Ex 1.10 - On mean and variance of two variables (Pattern Recognition and ML by Christopher Bishop)
Question: Answer:From the solutions to book
Using ChatGPT
If x and y are two independent continuous variables, can you derive the expression for E[x+y]Certainly! If x and y are two independent continuous random variables, you can find the expected value of their sum E[x+y] by using the property that for independent random variables, the expected value of the sum is the sum of the expected values. Mathematically:
E[x+y]=E[x]+E[y]
So, the expected value of the sum of two independent continuous random variables is simply the sum of their individual expected values.
If you have specific probability density functions (PDFs) for x and y, you can also find E[x+y] by integrating over all possible values of x and y:
E[x+y]=∫−∞∞∫−∞∞(x+y)⋅f(x)⋅g(y)dxdy
Here, f(x) and g(y) are the probability density functions of x and y respectively.
If x and y have known probability density functions, you can substitute them into the above expressions to find the expected value of their sum.
Certainly! Let's integrate the last expression and show that for independent and identically distributed (iid) random variables x and y, E[x+y]=E[x]+E[y].
Assuming f(x) and g(y) are the probability density functions (PDFs) of x and y respectively, and x and y are iid, the expression for E[x+y] is:
E[x+y]=∫−∞∞∫−∞∞(x+y)⋅f(x)⋅g(y)dxdy
Now, let's perform the integration:
E[x+y]=∫−∞∞∫−∞∞(x+y)⋅f(x)⋅g(y)dxdy
=∫−∞∞∫−∞∞x⋅f(x)⋅g(y)dxdy+∫−∞∞∫−∞∞y⋅f(x)⋅g(y)dxdy
Now, by the independence of x and y, the joint PDF f(x,y) can be expressed as the product of the individual PDFs:
f(x,y)=f(x)⋅g(y)
So, we can simplify the above expression:
=∫−∞∞x⋅f(x)(∫−∞∞g(y)dy)dx+∫−∞∞y(∫−∞∞f(x)dx)⋅g(y)dy
The integrals inside the parentheses evaluate to 1 because f(x) and g(y) are probability density functions, which integrate to 1 over their respective ranges. Therefore:
=∫−∞∞x⋅f(x)dx+∫−∞∞y⋅g(y)dy
This is equivalent to E[x]+E[y], showing that for iid x and y, E[x+y]=E[x]+E[y].