Suppose the random variable Y has a mean of 6969 and a variance of 6464. Let Z​ = StartFraction 1 Over StartRoot 64 EndRoot EndFraction
1
64​(Y dash– 6969​).
Part 2
Show that mu Subscript Upper Z Baseline equals 0μZ=0.
mu Subscript Upper Z equals Upper E left bracket nothing left parenthesis Upper Y minus nothing right parenthesis right bracketμZ=E enter your response here Y−enter your response here
     equals nothing left bracket mu Subscript Upper Y minus nothing right bracket equals 0=enter your response here μY−enter your response here=0
                                                                                                                                                                                                                                                                                          
​(Round your responses to two decimal places​)

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!

Suppose the random variable Y has a mean of 6969 and a variance of 6464. Let Z = (1 / sqrt(64)) * (Y - 6969). Show that μZ = 0. Round your responses to two decimal places.

Learn how to calculate the mean of the random variable Z, defined by the transformation Z = (1 / sqrt(64)) * (Y - 6969), where Y has a mean of 6969 and a variance of 6464. This problem explores key probability concepts, including expectation, variance, and the linearity of expectation. Suitable for advanced high school or college-level students.

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