Question

If \( X \) is a normal distribution with mean \( \mu \) and variation \( \sigma^2 \), then the standard deviation and the mean of \( Z = \frac{x - \mu}{\sigma} \) are ........... , ...........

• A. \( \sigma, \mu \)
• B. 1, 0
• C. 0, \( \frac{1}{2} \)
• D. \( \frac{1}{2}, 0 \)

Hint:

The transformation \( Z = \frac{x - \mu}{\sigma} \) standardizes a normal distribution, making the mean 0 and the standard deviation 1.

Answer 1

The Correct Option is B

The given problem involves the standardization of a normal distribution. The transformation \( Z = \frac{x - \mu}{\sigma} \) is used to standardize the random variable \( X \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.
- The mean of the standardized variable \( Z \) is 0, since subtracting \( \mu \) centers the distribution around 0.
- The standard deviation of \( Z \) is 1, since dividing by \( \sigma \) scales the distribution so that its standard deviation becomes 1.
Thus, the standard deviation of \( Z \) is 1, and its mean is 0. Therefore, the correct answer is \( 1, 0 \).