Question

the random variable ( x ) has mean 12 and standard deviation 3. the random variable ( w ) is defined as ( w = 7 + 2x ). what are the mean and standard deviation of ( w )?
a the mean is 24, and the standard deviation is 6.
b the mean is 24, and the standard deviation is 13.
c the mean is 31, and the standard deviation is 3.
d the mean is 31, and the standard deviation is 6.
e the mean is 31, and the standard deviation is 13.

Explanation:

Step1: Calculate mean of $W$

Use linearity of expectation: $E[W] = E[7+2X] = 7 + 2E[X]$
Substitute $E[X]=12$:
$E[W] = 7 + 2\times12 = 7 + 24 = 31$

Step2: Calculate standard deviation of $W$

For standard deviation: $\sigma_W = |2|\sigma_X$
Substitute $\sigma_X=3$:
$\sigma_W = 2\times3 = 6$

Answer:

D. The mean is 31, and the standard deviation is 6.