Question

What happens if the correlation is not zero? Suppose that X is a random variable with mean 20 and standard deviation 3. Also suppose that Y is a random variable with mean 60 and standard deviation 2 . Assume that the correlation between X and Y is 0.4. Find the variance and the standard deviation of the random variable Z for each of the following cases. Be sure to show your work. (a) Z=33−8X. (b) Z=11X−6 (c) Z=X+Y. (d) Z=X−Y. (e) Z=−2X+2Y.

Answer 1

To calculate the z-score for y=4 when Y has a mean of 2 and a standard deviation of 1, subtract the mean from 4 and divide by the standard deviation, resulting in a z-score of 2.

Step-by-step explanation:

To find the z-score for a given value, such as y = 4 in the context of the random variable Y with distribution Y~ N(2,1), we use the formula for calculating z-scores: z = (y - mean) / standard deviation. In this case, the mean (μ) is 2 and the standard deviation (σ) is 1.

The z-score for y = 4 would be calculated as follows:

z = (4 - 2) / 1

z = 2 / 1

z = 2

This means that a y value of 4 is exactly 2 standard deviations above the mean of the distribution Y. Therefore, both x = 17 and y = 4 are situated 2 standard deviations to the right of their respective means in their normal distributions.