Final answer:
The new expected value after the linear transformation 3x - 3 is 12 and the new variance is 18 for a random variable x with mean 5 and variance 2.
Step-by-step explanation:
According to the information provided, we are given a random variable x with a mean of 5 and a variance of 2.
When transforming this random variable using the linear transformation 3x - 3, we need to find the new expected value (E[3x - 3]) and the new variance (Var[3x - 3]).
The expected value of a linear transformation of a random variable is calculated by E[aX + b] = aE[X] + b, where a and b are constants. In this case, a = 3 and b = -3.
So, the new expected value is E[3x - 3] = 3E[x] - 3 = 3(5) - 3 = 15 - 3 = 12.
Regarding the variance, it is affected only by the scaling factor a, not by the shift b. Importantly, Var[aX + b] = a2Var[X].
Hence, the new variance is Var[3x - 3] = 32Var[x] = 9(2) = 18.