# Category Archives: Series

## Using a Mass Function

Let $$X$$ be a random variable with probability mass function

$$p(n) = \frac{1}{c^n}\quad \text{for } n=2,3,4,\cdots$$
and $$p(x)=0$$ otherwise.

1. Find $$c.$$
2. Compute the probability that $$X$$ is even.

## Magic Die (or is it rigged?)

A magician claims to have a magic die. If the die is rolled and lands on an even number, then the next time the die is rolled it will land on an odd number (and vice versa). So, as the die is rolled it will alternate perfectly between even and odd numbers (or so the magician claims).

You, being skeptical, figure there’s a 1 percent chance that the die is magical and a 99 percent chance that it’s just an ordinary fair die. You then ask the magician to “prove” the die is magical by rolling it some number of times.

How many successfully alternating rolls will it take for you to think there’s a 99 percent chance the die is magical (or, more likely, that it’s rigged in some way so it always alternates)?

## Flipping Coins and Independence

An experimenter has two fair coins and one biased coin. The biased coin lands on heads with probability 3/4.

The experimenter randomly selects one of the three coins and flips it until they get heads.

Let $$A$$ be the event that the experimenter flipped the biased coin.
Let $$B$$ be the event that it took the experimenter an even number of flips to get heads.

Are events $$A$$ and $$B$$ independent?

## Coin flipping game

Your friend challenges you to a game in which you flip a fair coin until you get heads. If you flip an even number of times, you win. Let $$A$$ be the event that you win. Let $$B$$ be the event that you flip the coin 3 or more times. Let $$C$$ be the event that you flip the coin 4 or more times.

1. Compute $$\mathbb{P}(A)$$.
2. Are $$A$$ and $$B$$ independent?
3. Are $$A$$ and $$C$$ independent?

## Taylor Series

Write in first 4 terms of the Taylor series expansion around $$x=0$$ for the following functions:

1. $$\log(1+x)$$
2. $$e^{a x}$$
3. $\frac{1}{1-x}$

## Arithmetic sum

Show that

$\sum_{j=1}^n j =\frac{1}{2}n(n+1)$

Hint: notice that $$1+n=n+1$$, $$2+(n-1)=n+1$$, $$3+(n-2)=n+1$$ and so on. How many such pairings exist ?