Distribution Convergence

Let’s do a problem from Chapter 5 of All of Statistics.

Suppose X_1, \dots X_n \sim \text{Uniform(0,1)} . Let Y_n = \bar{X_n}^2 . Find the limiting distribution of Y_n .

Note that we have Y_n = \bar{X_n}\bar{X_n}

Recall from Theorem 5.5(e) that if X_n \rightsquigarrow X and Y_n \rightsquigarrow c  then X_n Y_n \rightsquigarrow cX .

So the question becomes does X_n \rightsquigarrow c  so that we can use this theorem? The answer is yes. Recall that from Theorem 5.4(b) X_n \overset{P}{\longrightarrow} X implies that X_n \rightsquigarrow X . So if we can show that we converge to a constant in probability we know that we converge to the constant in distribution. Let’s show that \bar{X}_n \overset{P}{\longrightarrow} c . That’s easy. The law of large numbers tells us that the sample average converges in probability to the expectation. In other words \bar{X}_n \overset{P}{\longrightarrow} \mathbb{E}[X] . Since we are told that X_i is i.i.d from a Uniform(0,1) we know the expectation is \mathbb{E}[X] = .5 .

Putting it all together we have that:

Y_n = \bar{X_n}^2
Y_n = \bar{X_n}\bar{X_n}
Y_n \rightsquigarrow \mathbb{E}[X]\mathbb{E}[X] (through the argument above)
Y_n \rightsquigarrow (.5)(.5)
Y_n \rightsquigarrow .25

We can also show this by simulation in R, which produces this chart:


Indeed we also get the answer 0.25. Here is the R code used to produce the chart above:

# Load plotting libraries

# Create Y = g(x_n)
g = function(n) {

# Define variables
n = 1:10000
Y = sapply(n, g)

# Plot
df = data.frame(n,Y)
ggplot(df, aes(n,Y)) +
  geom_line(color='#3498DB') +
  theme_fivethirtyeight() +
  ggtitle('Distribution Convergence of Y as n Increases')

You Should Literally Read This

Merriam Webster has a fantastic entry on the use of the word “literally.” Here is the introduction; there is much more of interest at the link.

Is it ever okay to use literally to mean “figuratively”?

F. Scott Fitzgerald did it (“He literally glowed”). So did James Joyce (“Lily, the caretaker’s daughter, was literally run off her feet”), W. M. Thackeray (“I literally blazed with wit”), Charlotte Brontë (“she took me to herself, and proceeded literally to suffocate me with her unrestrained spirits”) and others of their ilk.

Solitary Confinement

According to the law, deprivation of freedom alone is supposed to be the price society exacts for crimes. Even within this mostly punitive model, people are supposed to be sent to prison as punishment, not for punishment.

That is from an article from Minnesota’s Start Tribune by an inmate that spent 585 days in solitary confinement. It does not sound pleasant.

Imagine being locked in a concrete room the size of your bathroom for 20 months with no way out. Under the glare of bright fluorescent lights that never go dark, the only way to tell day from night is by what type of meal slides through a hole in the door.

Now imagine that door is soundproof and the only noises you’ve heard for almost two years are your own voice and the occasional faint metallic banging as someone loses his mind in another room near yours. Imagine being so deprived of stimulation that watching ants race to a chunk of cookie for hours was the most exciting event of those nearly 600 days.

What you are imagining was my life.

In fact the Star Tribune has a related 4-part series called “Way Down in the Hole,” which I hope to read soon.

Death by A.I.

If you want a picture of A.I. gone wrong, don’t imagine marching humanoid robots with glowing red eyes. Imagine tiny invisible synthetic bacteria made of diamond, with tiny onboard computers, hiding inside your bloodstream and everyone else’s. And then, simultaneously, they release one microgram of botulinum toxin. Everyone just falls over dead.

That is from a new profile of Elon Musk and other A.I. critics and proponents in Vanity Fair.

The Phone Romeos of India

That is the topic of a short, fascinating new article in The New York Times.

The “phone Romeo,” as he is known here, calls numbers at random until he hears a woman’s voice, in the hope of striking up a romantic attachment. Among them are overeager suitors (“Can I recharge your mobile?”), tremulous supplicants (“I am talking to you, madam, but my body is shaking”) and the occasional heavy breather (“I want to do the illegal things with you”).

Intentionally dialing wrong numbers is a labor-intensive way to find a girlfriend. But it is increasingly common in a range of countries — Morocco, Papua New Guinea, Bangladesh and India are examples — where traditional gender segregation has collided head-on with a wave of cheap new technology.

At the police call center in Lucknow, in northern India, roughly 700 calls come in every day, mostly from women complaining of persistent calls from strange men. The Hindustan Times recently reported that phone recharging outlets were selling the numbers of young women to interested men, charging 500 rupees, about $7.60, for a “beautiful” girl and 50 rupees for an “ordinary” one.

In related impacts of technology on love, some experts believe Tinder is decreasing the importance of home court advantage in the NBA by making hookups more efficient (no need to spend all night at the club before an away game).

Notes on Big Tobacco

Tim Harford has a new article called “The Problem with Facts.” Here is one very interesting bit.

Prusiner is a neurologist. In 1972, he was a young researcher who’d just encountered a patient suffering from Creutzfeldt-Jakob disease. It was a dreadful degenerative condition then thought to be caused by a slow-acting virus. After many years of study, Prusiner concluded that the disease was caused instead, unprecedentedly, by a kind of rogue protein. The idea seemed absurd to most experts at the time, and Prusiner’s career began to founder. Promotions and research grants dried up. But Prusiner received a source of private-sector funding that enabled him to continue his work. He was eventually vindicated in the most spectacular way possible: with a Nobel Prize in Medicine in 1997. In his autobiographical essay on the Nobel Prize website, Prusiner thanked his private-sector benefactors for their “crucial” support: RJ Reynolds, maker of Camel cigarettes.

The tobacco industry was a generous source of research funds, and Prusiner wasn’t the only scientist to receive both tobacco funding and a Nobel Prize. Proctor reckons at least 10 Nobel laureates are in that position. To be clear, this wasn’t an attempt at bribery. In Proctor’s view, it was far more subtle. “The tobacco industry was the leading funder of research into genetics, viruses, immunology, air pollution,” says Proctor. Almost anything, in short, except tobacco. “It was a massive ‘distraction research’ project.” The funding helped position Big Tobacco as a public-spirited industry but Proctor considers its main purpose was to produce interesting new speculative science. Creutzfeldt-Jakob disease may be rare, but it was exciting news. Smoking-related diseases such as lung cancer and heart disease aren’t news at all.

Will Trump Maintain Republican Support?

Jeet Heer writing for New Republic thinks yes:

These numbers make clear why a large chuck of Trump’s base won’t object to a budget that punitively targets the poor. Most of his voters are not poor, but regular old Republicans, for whom the budget released Thursday is just the latest version of policies they’ve supported since the Reagan era.

To be sure, some of Trump’s white working class supporters might become disillusioned once they realize the consequences of his policies. But it’s just as likely that they’ll be pleased, as will Trump’s larger coalition, by the restrictionist and isolationist aspects of his agenda, like his targeting of Muslim immigrants and plan to build a wall along the Mexican border.

Trump’s gamble, if indeed this is a witting strategy, is that he can hold his base together on a shared support for ethno-nationalism, with the bulk of economic benefits going to well-to-do Republicans. So far, to judge by the even keel of his poll numbers, his bet is working. This might change when Trump’s economic policies move from the realm of proposals to actual policies that affect everyday lives. But for now, the argument that Trump is betraying his base is just a rhetorical meme that has little bearing on how his voters feel about him.