Final answer:
The variance of a new random variable created by a linear transformation, var(2x + 1), when x has a mean of 2 and a variance of 3, is computed using the properties of variance and the calculation results in a variance of 12.
Step-by-step explanation:
The student has asked for help in computing the variance of a new random variable that is a linear transformation of another random variable with a given mean and variance. Specifically, the question is about finding var(2x + 1) when the mean of the random variable x is 2 and the variance is 3.
To compute the variance of a transformed random variable, we use the properties of variance. For a linear transformation of the form Y = aX + b, where X is a random variable and a and b are constants, the variance of Y is given by Var(Y) = a² • Var(X). The mean of the new random variable is not needed for the calculation of the variance.
Given that X has a variance of 3, the variance of 2X would be (2²) • 3 which equals 4 • 3 = 12. Since the variance of a constant is 0, adding 1 does not affect the variance. Therefore, Var(2X + 1) = 12.