create an example of i.i.d Bernoulli random variable

Question

i need to create an example of identically distributed, but dependent Bernoulli random variables where $ x_1,x_2....x_n$ i.e $x\in{0,1}$ such that, $$P\big(|\mu-\frac{1}{n}\sum_{i=1}^{n}x_i|\geq \frac{1}{2}\big)=1$$

where $\mu=E[x_i]$,

The example should show that independence is crucial for convergence of mean to the expected values. i.e $\mu=E[x_i]$

im a non mathematics student struglling a bit with understanding the concept.i was wondering how $\mu$ equals $E[x_i]$ and its relation to independence. a short explanation will really help.

Answer 1

Take $(x_1,...,x_n)$ such that $x_1$ is a Bernoulli r.v. with parameter $1/2$, and $x_1=x_2=\dots=x_n$ a.s. Then $\mu=1/2$, but $$ \frac{1}{n}\sum_{i=1}^n x_i = x_1\in\{0,1\} $$ so $ \left\lvert \mu - \frac{1}{n}\sum_{i=1}^n x_i \right\rvert = 1/2 $ a.s.

Answer 2

Let $x_1$ be chosen at random (with $\mu=0.5$) and $x_{k+1}=1-x_k$ for all other $k$. These are obviously dependent and satisfy the condition on $P$.

Answer 3

Since we need $P \left(|\mu − \frac{1}{n}\sum_{i=1}^n x_i| \geq a \right)=1$, here $a = \frac{1}{2}$, then as long as our $(x_1, ..., x_n)$ such that $x_1$ is a Bernoulli rv. then we just need the parameter of the bernoulli distribution to be $\geq a$.

Which means $|\mu - \frac{1}{n}\sum_{i=1}^n x_i|\geq a$.

Therefore, in this case, we can choose $x_1$ to be bernoulli distributed with parameter equal to or greater than $\frac{1}{2}$.

Hope this helps any future questions for students in the same situation.