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Let $X$ be a discrete uniform random variable on the set $\{000, 011, 101, 110\}$ of four binary integers, and let $X_{i}$ denote the ith digit of $X$, for $i = 1, 2, 3$. Show that $X_{1}, X_{2}, X_{3}$ are independent pairwise, but not totally independent. Can you generalize this example to more than three random variables?

Can anyone help me with this exercise?

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In order to help you with homework, you need to show us what you've done so far. – Ayman Hourieh Apr 29 '12 at 22:59
For example, do you know what pairwise independent means? Do you know what "totally independent" means? – Gerry Myerson Apr 29 '12 at 23:09
Also posted on stats.SE by a different user. – Dilip Sarwate Apr 30 '12 at 12:11
up vote 3 down vote accepted

Here is a construction that I find worth knowing.

Let $n\geqslant2$ and $K_n=\{0,1\}^n$ denote a discrete hypercube of dimension $n$. Assume that the random vector $Y^{(n)}=(Y_i)_{1\leqslant i\leqslant n}$ is uniformly distributed on $K_n$, hence $(Y_i)_{1\leqslant i\leqslant n}$ is a family of i.i.d. uniform Bernoulli random variables. For any proper subset $I$ of $\{1,2,\ldots,n\}$, let $Y^I=(Y_i)_{i\in I}$ and $K_I=\{0,1\}^I$. And now, the key part:

Let $L_n\subset K_n$ denote the set of points $(x_i)_{1\leqslant i\leqslant n}$ in $K_n$ such that $\sum\limits_{i=1}^nx_i$ is even.

Assume first that $n=3$.

  • Show that, conditionally on $[Y^{(3)}\in L_3]$, $Y^{(3)}$ is distributed like $(X_1,X_2,X_3)$ in your homework.

For every $n\geqslant2$, show the following:

  • The random vector $Y^I$ is uniformly distributed on $K_I$. Equivalently, $Y^I$ is i.i.d. uniform Bernoulli.
  • Conditionally on $[Y^{(n)}\in L_n]$, $Y^{(n)}$ is not independent.
  • Conditionally on $[Y^{(n)}\in L_n]$, $Y^I$ is uniformly distributed on $K_I$. Equivalently, conditionally on $[Y^{(n)}\in L_n]$, $Y^I$ is i.i.d. uniform Bernoulli.

For example:

Conditionally on $[Y^{(n)}\in L_n]$, $Y^{(n)}$ is not independent but any $n-1$ of its coordinates are independent (and distributed uniformly on $K_{n-1}$).

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