We are given the random variable $Y$ with a mean $\mu_Y = 6969$ and variance $\sigma_Y^2 = 6464$. The variable $Z$ is defined as:

$Z = \frac{1}{\sqrt{64}} (Y - 6969)$

Part 1: Show that $\mu_Z = 0$

We need to find the mean $\mu_Z$ of the random variable $Z$.

By the definition of the expectation (mean), we have:

$\mu_Z = E[Z]$

Substitute the expression for $Z$:

$\mu_Z = E\left[\frac{1}{\sqrt{64}} (Y - 6969)\right]$

Since the expectation is linear, we can separate it:

$\mu_Z = \frac{1}{\sqrt{64}} E[Y - 6969]$

Now, simplify $E[Y - 6969]$:

[ E[Y - 6969] = E[Y] - 6969 = \mu_Y - 6969 = 6969 - 6969 = 0 ]

Thus,

$\mu_Z = \frac{1}{\sqrt{64}} \times 0 = 0$

So, $\mu_Z = 0$.

Summary:

We have shown that the mean of $Z$ is indeed 0.

Would you like more details, or do you have any questions about this?

Here are five related questions:

1. How do you compute the variance of a transformed random variable?
2. What properties of expectation were used in this calculation?
3. How would the result change if $Z$ had a different scaling factor?
4. Can we find the standard deviation of $Z$ from the given information?
5. How does linearity of expectation apply in more complex cases?

Tip: Remember, the expectation (mean) of a constant is just the constant itself!