Final answer:
To find the new mean and standard deviation after transforming a random variable, you apply the arithmetic operations to the mean and multiply the standard deviation by the coefficient of the variable if there's a multiplication. The mean of y=x+6 is 3, v=9x is -27, and w=9x+6 is -21. The standard deviation remains 2 for y and becomes 18 for both v and w.
Step-by-step explanation:
The student is asking about finding the mean and standard deviation of new random variables that are linear transformations of the original random variable x.
- For y = x + 6, the new mean is the original mean of x (-3) plus 6, which equals 3. The standard deviation does not change with addition or subtraction, so the standard deviation of y is still 2.
- For v = 9x, the new mean is 9 times the mean of x, that is 9 * (-3) = -27. The new standard deviation is also 9 times the original standard deviation of x, so the standard deviation of v is 9 * 2 = 18.
- For w = 9x + 6, the mean of w would be the same as the mean of v (calculated above) plus 6, so the mean of w is -27 + 6 = -21. The standard deviation of w is the same as the standard deviation of v, because addition does not affect standard deviation, hence it remains 18.