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Unit 02 /
Problem 3:


I independent random variables
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Easy


(Can Independence Carry Through?)

The following statements are either true or false. Prove them or provide a counterexample:

  1. If X, Y, and Z are random variables, X and Y are independent, Y and Z are independent, then X and Z must be independent.

  2. If X, Y, and Z are random variables, X and Y are not independent, Y and Z are not independent, then X and Z must be not independent.

Answer:
  1. False, 2. False
Solution:

1. Proof by counterexample:

Imagine that we have $2$ fair coins, coin A and coin B. We define several random variables based on the outcome of these coins.

  • $X$ takes on value of $0$ for tails in coin A, $1$ for heads in coin A
  • $Y$ takes on value of $0$ for tails in coin B, $1$ for heads in coin B
  • $Z$ takes on value of $0$ for tails in coin A, $1$ for heads in coin A

It can be seen that $X$ and $Z$ will always have exactly the same outcome. They are therefore not indpendentHowever, we can also see that since coin A is indpendent of coin B, $X \perp Y$ and $Y \perp Z$.

Therefore supposition 1 is False

2. Proof by counterexample:

We can imagine we have $2$ fair coins, coin A and coin B

From these $2$ coins we define $3$ random variables

  • $X$ takes on value of $0$ for tails in coin A, $1$ for heads in coin A
  • $Z$ takes on value of $0$ for tails in coin B, $1$ for heads in coin B
  • $Y$ takes on the value of the sum of $X$ and $Y$

By inspection $X$ and $Z$ are independent of each other. $Y$ and $X$ are not independent niether are $Y$ and $Z$.

Therefore supposition 2 is False