Chapter 4 Random Variables
Ver.102525
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What is a random variable ?
A formal definition :
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Example
Therefore,
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Discrete Random Variables
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An example on the random variable value v.s. probability
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Example
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Cumulative distribution function (cdf)
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Cumulative distribution function (cdf)
for a discrete random variable.
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Expected value
Example
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Example
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Suppose the random variable X is nonnegative and integer-valued, then
This can be seen by observing that
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Expectation of a function of a random variable
Attention !!
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Example
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Variance
Alternatively,
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Example
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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�person’s household
3
4⅓
A general solution
This little mobius strip of a phenomenon is called the “generalized friendship paradox,” and at first glance it makes no sense. Everyone’s friends can’t be richer and more popular — that would just escalate until everyone’s 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. Here’s how MIT’s 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.
It’s 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. That’s the reason that, on average, your friends have more friends than you do.
And this rule doesn’t 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 you’ve 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 they’re set in real life or online.
So while it’s 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 people’s 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.
That’s 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
It can be shown that E[Y] ≥ E[X] in any social network . To see this, note that for a connected simple k-vertex graph, we have
Alice
where di is the valency of vertex i.
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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 Cauchy’s inequality
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Bernoulli distribution
Binomial distribution
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Example
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Properties
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As a result,
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Poisson distribution
It can be derived from the binomial distribution, under some extreme conditions…
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size
We have
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Note that
As a result,
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In the birthday problem,
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A closer look at Poisson distribution:
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As a result,
and
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Occurrence rate and waiting time
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Other property
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Geometric distribution
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As a result,
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As a result,
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Negative binomial
success.
SHOW THAT
It is well known that
?
We have more!
It is well known that
We have more!
HINTs
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Back to the problem
?
pf.
Since negative binomial is involved, the distribution is called the Negative binomial distribution.
Suppose
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Using negative binomial to prove the Chu-Vandermonde identity:
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Example
Compared to the solution by Fermat in Chapter 3, we have an interesting identity:
We show this in the following.
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We need the identity below.
To see this, we check on both sides of the equality the coefficient of , p>n. In the left hand side we have
which is the same as that in the right hand side. (Here we
recall from Ch.1 that)
Letting a=p and b=1-p yields the result in the previous page.
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The general playoff problem in Ch.3.
Suppose team A and B meet for a (2k+1)-game playoff for some positive integer k. They play until one team wins k+1 games. If team A has a winning probability p in every game, and Pi is the probability that i games are played, show that
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We first prove that
thus the proof is completed.
pf.
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The probability that i games are played is
Finally, we have
where follows from the
previous page by letting a=p and b=1-p.
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Hypergeometric distribution
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ment
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Note that
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Expected values of Sums of random variables.
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Example
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More generally,
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Properties of cumulative distribution function:
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Example
We thus have
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Let q=1-p.
and the proof is completed.
pf:
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As a result,