PA-4 / Lesson 3
Problem 1.
You roll two fair dice. Let $M$ be a random variable representing the minimum of the numbers that you roll. Find the probability distribution and the maths expectation of $M$.
(Remark for those who know too much: we have not proven any properties of $\mathbb{E}$, in particular the linearity of it. Thus please stick to the first principles.)
Problem 2.
Let $(\Omega, \mathbb{P})$ be a finite probability space and assume there are two random variables defined on this probability space, say $X$ and $Y$. Is it necessarily true that $\mathbb{E}[X \cdot Y] = \mathbb{E} [X] \cdot \mathbb{E}[Y]$?
Problem 3.
A bus ticket costs 3 pounds and the penalty fare for not having a ticket is 40 pounds. If $p$ is the probability you meet a ticket inspector, for which $p$ it is more profitable (in expectation) not to buy ticket?
Problem 4.
In a box there are 2009 socks which are blue or red. Is it possible for there to be some number of blue socks so that the probability of fishing out two socks of the same colour is $\frac{1}{2}$?
Problem 5.
Find the mathematical expectation of a binomial random variable with parameters $n$ and $p$.
(Remark for those who know too much: please avoid using the linearity of expectation or analogous properties. Try working from the first principles and performing smart algebraic manipulations to get to the result).
Interestingly, the word "random" did not always mean what we now claim it means. It's etymology (i.e the study of the origins and the way in which the meanings have changed throughout history) is in fact quite rich.
First attested in the early 1300s, random originally referred to “great speed” or “force”, used especially in the phrases to run at random or with great random. Random’s velocity and violence conveyed a sense of impetuousness and rashness. And so by the 1540s, the expression at random rushed towards “without aim or purpose,” a short step from the modern adjective, which settled in by the 1650s.
Random did not come into English at random. It derives from the Old French randon, a similar noun denoting “speed, haste, violence, impetuousness,” probably formed from the verb randir, “to run fast, gallop.” The deeper origins of randir aren’t certain, but scholars conjecture the Frankish *rant, “a running.” (Frankish was a West Germanic language spoken by the Franks, a Germanic tribe in late antiquity whose name lives on in France and the adjective frank.) This *rant, and thus random, may be related to the same Germanic root that gives us run.
The statistical random emerges by the 1880s, but the word was not done running. Computing, campus, and teen slang in the 1960–80s helped fashion a random, a “stranger” or “outsider,” sometimes shortened to rando, as well as the informal, often pejorative use of random for “odd, peculiar, unexpected, unfamiliar,” e.g., Don’t go home with that random guy. Definitely run–at the etymological random–from him.