Question 1
Consider the following:
- You have a random variable $X$, which is just a number (or a set of numbers)
that is randomly chosen. In this case, $X$ is a string of n bits (where each bit
is either 0 or 1), and each bit is chosen completely at random.
- You also have another random variable $Y$, which is also an n-bit string.
However, $Y$ doesn’t have to be random in the same way as $X$—it can follow any
pattern. But $X$ and $Y$ are independent, meaning knowing $Y$ doesn’t tell us
anything about $X$.
- Now, we define a new random variable $Z$ by taking $X$ and $Y$ and combining them
using something called the ”bitwise XOR operation” (written as $\oplus$). XOR works
like this for each bit:
– If the two bits are the same (both 0 or both 1), the result is 0.
– If the two bits are different (one is 0 and the other is 1), the result
is 1.
So, assuming we define $Z = X \oplus Y$, we apply this rule to each bit in the string
separately.