Prologue to Measurements

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We get a kick out of the chance to believe that we have control over our lives. In any case, actually there are numerous things that are outside our control.Everyday we are defied by our own particular ignorance.According to Albert Einstein:

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Prologue to Statistics Dr. P Murphy

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Why examine Statistics? We jump at the chance to feel that we have control over our lives. Be that as it may, in all actuality there are numerous things that are outside our control. Regular we are stood up to by our own particular numbness. As per Albert Einstein: "God does not play dice." But rather we as a whole ought to know superior to Prof. Einstein. The world is represented by Quantum Mechanics where Probability rules.

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Consider a typical day for a normal UCD understudy. You get up in the morning and the daylight hits your eyes. At that point all of a sudden without notice the world turns into an unverifiable place. To what extent will you need to sit tight for the Number 10 Bus toward the beginning of today? When it arrives will it be full? Will it be out of administration? Will it be sprinkling while you hold up? Will you be late for your 9am Maths address?

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Probability is the Science of Uncertainty . It is utilized by Physicists to foresee the conduct of basic particles. It is utilized by designers to fabricate PCs. It is utilized by business analysts to foresee the conduct of the economy. It is utilized by stockbrokers to profit on the stockmarket. It is utilized by analysts to figure out whether you ought to land that position.

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What about Statistics? Measurements is the Science of Data. The Statistics you have seen before has been presumably been Descriptive Statistics. Furthermore, Descriptive Statistics made you feel like this … .

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What is Inferential Statistics? It is a teach that permits us to gauge obscure amounts by making some rudimentary estimations. Utilizing these appraisals we can then make Predictions and Forecast the Future A Crystal Ball

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Chapter 1 Probability

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Consider a Real Problem Can you profit playing the Lottery? Give us a chance to figure odds of winning. To do this we have to take in some essential standards about likelihood. These standards are for the most part only methods for formalizing essential judgment skills . Case: What are the odds that you get a HEAD when you flip a coin? Case: What are the odds you get a joined aggregate of 7 when you move two dice?

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1.1 Experiments An E xperiment prompts to a solitary result which can't be anticipated with conviction. Illustrations Toss a coin : head or tail Roll a kick the bucket : 1, 2, 3, 4, 5, 6 Take pharmaceutical : more awful, same, better Set of all results - S plentiful S pace . Flip a coin Sample space = {h,t} Roll a kick the bucket Sample space = {1, 2, 3, 4, 5, 6} 

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1.2 Probability The P robability of a n result is a number somewhere around 0 and 1 that measures the probability that the result will happen when the trial is performed. (0=impossible, 1=certain). Probabilities of all example indicates must entirety 1. Long run relative recurrence translation. Illustration: Coin hurling test P(H)=0.5 P(T)=0.5

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1.3 Events An occasion is a particular accumulation of test focuses. The likelihood of an occasion An is figured by summing the probabilities of the results in the example space for A.

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1.4 Steps for ascertaining Probailities Define the test. List the example focuses. Dole out probabilities to the specimen focuses. Decide the gathering of test focuses contained in case of intrigue. Whole the example guide probabilities toward get the occasion likelihood.

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Example: THE GAME Of CRAPS In Craps one moves two reasonable dice. What is the likelihood of the aggregate of the two dice indicating 7?

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1.5 Equally likely results So the Probability of 7 when moving two dice is 1/6 This case shows the accompanying standard: In a Sample Space S of similarly likely results. The likelihood of the occasion An is given by P(A) = #A/#S That is the quantity of results in A partitioned by the aggregate number of occasions in S.

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1.6 Sets A compound occasion is a creation of at least two different occasions. A C : The Complement of An is the occasion that A does not happen A  B : The U nion of two occasions An and B is the occasion that happens if either An or B or both happen , it comprises of all example indicates that have a place An or B or both. A  B : The I ntersection of two occasions An and B is the occasion that happens if both An and B happen , it comprises of all example indicates that have a place both An and B

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1.7 Basic Probability Rules P(A c )=1-P(A) P( A  B )=P(A)+P(B)- P( A  B ) Mutually Exclusive Events will be occasions which can't happen in the meantime. P ( A  B )=0 for Mutually Exclusive Events.

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1.8 Conditional Probability P(A | B) ~ Probability of An occuring given that B has happened. P(A | B) = P ( A  B )/P(B) Multiplicative Rule: P ( A  B ) = P(A|B)P(B) = P(B|A)P(A)

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1.9 Independent Events An and B are free occasions if the event of one occasion does not influence the likelihood of the othe occasion. In the event that An and B are autonomous then P(A|B)=P(A) P(B|A)=P(B) P ( A  B )=P(A)P(B)

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Chapter 1 Probability EXAMPLES

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Probability as an issue of life and passing

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Positive Test for Disease 1 in each 10000 individuals in Ireland experience the ill effects of AIDS There is a test for HIV/AIDS which is 95% precise. You are not feeling great and you go to doctor's facility where your Physician tests you. He says you are sure for AIDS and lets you know that you have year and a half to live. In what capacity would it be a good idea for you to respond?

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Positive Test for Disease Let D be the occasion that you have AIDS Let T be the occasion that you test positive for AIDS P(D)=0.0001 P(T|D)=0.95 P(D|T)=?

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Positive Test for Disease

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Chapter 1 Examples Example 1.1 S={A,B,C} P(A) = ½ P(B) = 1/3 P(C) = 1/6 What is P({A,B})? What is P({A,B,C})? List all occasions Q with the end goal that P(Q) = ½.

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Chapter 1 Examples Example 1.2 Suppose that a speaker arrives late to class 10% of the time, leaves mid 20% of the time and both arrives late AND leaves mid 5% of the time. On a given day what is the likelihood that on a given day that teacher will either arrive late or leave early?

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Chapter 1 Examples Example 1.3 Suppose you are managed 5 cards from a deck of 52 playing cards. Discover the likelihood of the accompanying occasions 1. Every one of the four aces and the ruler of spades 2. Each of the 5 cards are spades 3. Every one of the 5 cards are diverse 4. A Full House (3 same, 2 same)

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Chapter 1 Examples Example 1.4 The Birthday Problem Suppose there are N individuals in a room. How huge ought to N be so that there is a more than half possibility that no less than two individuals in the room have a similar birthday?

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Chapter 1 Examples Example 1.4 Children are conceived similarly likely as Boys or Girls My sibling has two kids (not twins) One of his kids is a kid named Luke What is the likelihood that his other youngster is a young lady?

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Example 1.5 The Monty Hall Problem Game Show 3 entryways 1 Car & 2 Goats You pick an entryway - e.g. #1 Host knows what's behind every one of the entryways and he opens another entryway, say #3, and demonstrates to you a goat He then inquires as to whether you need to stay with your unique decision #1, or change to entryway #2?

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Ask Marilyn. Parade Magazine Sept 9 1990 Marilyn vos Savant Guinness Book of Records - Highest IQ "Yes you ought to switch. The principal entryway has a 1/3 shot of winning while the second has a 2/3 possibility of winning." Ph.D.s - Now two entryways, 1 goat & 1 auto so odds of winning are 1/2 for entryway #1 and 1/2 for entryway #2. "You are the goat" - Western State University.

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Who's privilege? Toward the begin, the example space is: { C GG, G CG, G GC } Pick an entryway e.g. #1 1 in 3 shot of winning Host demonstrates to you a goat so now { C G , G C G , G C } So Marilyn was correct, you ought to switch.

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Not persuaded? Envision a diversion with 100 entryways. 1 F430 Ferrari, 99 Goats. You pick an entryway. Have opens 98 of the 99 different entryways. Do you stay with your unique decision? Prob = 1/100 Or move to the unopened entryway. Prob = 99/100

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Boys, Girls and Monty Hall Sample Space ( posting most seasoned kid first) {GG, BG, GB, BB} Equally likely occasions One youngster is a boy: GG is inconceivable {BG, GB, BB} => P(OC = G) = 2/3 Luke is 6 months old. {GB, BB} => P(OC = G) = 1/2

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Odd Socks It is winter and the ESB are on strike. Toward the beginning of today when you woke up it was dim. In your sock drawer there was one sets of two dark socks and one odd cocoa one. Is it true that you are pretty much prone to wear coordinating socks today?

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EXAMS Seeing this confirmation amale understudy prosecutes UCD saying there is discimination against male understudies. UCD assembles all it's exam data together and reports the accompanying.

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Overall Female pass rate is 56% Overall Male pass rate is 60% HOW CAN THIS BE? Unmistakably UCD are LYING ! EXAM Pass Rates

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Overall Female pass rate is 56% Overall Male pass rate is 60% Simpson's Paradox

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Once upon a period in Hicksville, USA there was an evening time attempt at manslaughter mishap including a taxi. There are two taxi organizations in Hicksville, Green and Blue. 85% of cabs are Green and 15% are Blue. A witness recognized the taxi as being Blue. In the resulting court case the judge requested that the witness' perception under the conditions that won that night be tried. The witness accurately distinguished every shade of taxi 80% of the time. Attempt at manslaughter

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What is the likelihood that it was without a doubt a blue taxi that was included in the mischance? Attempt at manslaughter

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You are occasion in Belfast and a blast crushes the Odessey field. You are seen running from the blast and are captured. You are hence accused of being an individual from an endorsed paramilitary association and with bringing about the blast. In court you challenge your honesty. However the PSNI have DNA prove they assert joins you to the wrongdoing. DNA

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Their legal researcher conveys the accompanying crucial confirmation. The scientific researcher shows