Ver Chapter 4 Random Variables 1 SPHsu_Probbability 2 What is a random variable ? A formal definition : SPHsu_Probbability 3 Example Therefore, SPHsu_Probbability 4 Discrete Random Variables SPHsu_Probbability 5 An example on the random variable value v.s. probability SPHsu_Probbability 6 Example SPHsu_Probbability 7 Cumulative distribution function (cdf) SPHsu_Probbability 8 Cumulative distribution function (cdf) for a discrete random variable. SPHsu_Probbability 9 Expected value Example SPHsu_Probbability 10 Example SPHsu_Probbability 11 Suppose the random variable X is nonnegative and integer-valued, then This can be seen by observing that SPHsu_Probbability 12 SPHsu_Probbability 13 Expectation of a function of a random variable Attention !! SPHsu_Probbability 14 SPHsu_Probbability 15 Example SPHsu_Probbability 16 SPHsu_Probbability 17 SPHsu_Probbability 18 Variance Alternatively, SPHsu_Probbability 19 Example SPHsu_Probbability 20 pf. Proposition Average household size In 2011 the average household in Hong Kong had 2.9 people. Take a random person. What is the average number of people in his/her household? B: 2.9 A: < 2.9 C: > 2.9 Average household size average household size 3 3 average size of random persons household 3 44 A general solution This little mobius strip of a phenomenon is called the generalized friendship paradox, and at first glance it makes no sense. Everyones friends cant be richer and more popular that would just escalate until everyones a socialite billionaire. The whole thing turns on averages, though. Most people have small numbers of friends and, apparently, moderate levels of wealth and happiness. A few people have buckets of friends and money and are (as a result?) wildly happy. When you take the two groups together, the really obnoxiously lucky people skew the numbers for the rest of us. Heres how MITs Technology Review explains the math: The paradox arises because numbers of friends people have are distributed in a way that follows a power law rather than an ordinary linear relationship. So most people have a few friends while a small number of people have lots of friends. Its this second small group that causes the paradox. People with lots of friends are more likely to number among your friends in the first place. And when they do, they significantly raise the average number of friends that your friends have. Thats the reason that, on average, your friends have more friends than you do. And this rule doesnt just apply to friendship other studies have shown that your Twitter followers have more followers than you, and your sexual partners have more partners than youve had. This latest study, by Young-Ho Eom at the University of Toulouse and Hang-Hyun Jo at Aalto University in Finland, centered on citations and coauthors in scientific journals. Essentially, the generalized friendship paradox applies to all interpersonal networks, regardless of whether theyre set in real life or online. So while its tempting to blame social media for what the New York Times last month called the agony of Instagram that peculiar mix of jealousy and insecurity that accompanies any glimpse into other peoples glamorously Hudson-ed lives the evidence suggests that Instagram actually has little to do with it. Whenever we interact with other people, we glimpse lives far more glamorous than our own. Thats not exactly a comforting thought, but it should assuage your FOMO next time you scroll through your Facebook feed. Bob Mark Zoe Eve Sam Jessica X = number of friends Y = number of friends of a friend In your homework you will show that E[Y] E[X] in any social network Alice 26 SPHsu_Probbability Which way is more probable to find a firstborn child ? A random selection from the people in the street or a random call to any family to find ? A random selection from the people in the street: A random call to any family: By Cauchys inequality Bernoulli distribution Binomial distribution 27 SPHsu_Probbability Example 28 SPHsu_Probbability 29 SPHsu_Probbability 30 SPHsu_Probbability Properties 31 SPHsu_Probbability As a result, 32 SPHsu_Probbability 33 SPHsu_Probbability 34 SPHsu_Probbability Poisson distribution It can be derived from the binomial distribution, under some extreme conditions 35 SPHsu_Probbability size We have 36 SPHsu_Probbability Note that As a result, 37 SPHsu_Probbability In the birthday problem, 38 SPHsu_Probbability 39 SPHsu_Probbability 40 SPHsu_Probbability A closer look at Poisson distribution: 41 SPHsu_Probbability 42 SPHsu_Probbability 43 SPHsu_Probbability As a result, and 44 SPHsu_Probbability Occurrence rate and waiting time 45 SPHsu_Probbability Other property 46 SPHsu_Probbability Geometric distribution 47 SPHsu_Probbability As a result, 48 SPHsu_Probbability As a result, 49 SPHsu_Probbability Negative binomial success. Try to show that 50 SPHsu_Probbability Example 51 SPHsu_Probbability 52 SPHsu_Probbability Hypergeometric distribution 53 SPHsu_Probbability 54 SPHsu_Probbability 55 SPHsu_Probbability ment 56 SPHsu_Probbability Note that 57 SPHsu_Probbability 58 Expected values of Sums of random variables. SPHsu_Probbability 59 Example SPHsu_Probbability 60 More generally, SPHsu_Probbability 61 Properties of cumulative distribution function: SPHsu_Probbability 62 Example We thus have SPHsu_Probbability 63 as n - i is even. If n, i are both odd, we can let j=n-i, Let q=1-p. and the proof is completed. pf: SPHsu_Probbability 64 As a result,