Question

Let X₁ and X₂ be independent random variables. Suppose the mean and standard deviation of X₁ are 10 and 3, respectively, and the mean and standard deviation of X₂ are 12 and 4, respectively. If T = 2X₁ + 3X2, what is the variance of T?

A) 13
B) 48
C) 66
D) 180
E) 324

Answer 1

The variance of T, which is 2X₁ + 3X₂, where X₁ and X₂ are independent random variables with given means and standard deviations, is calculated using the formula for the variance of a sum of independent random variables. The correct answer is 180, which is option D.

Step-by-step explanation:

The question is about finding the variance of a new random variable T, which is a linear combination of two independent random variables X₁ and X₂. The formula for the variance of a sum of independent random variables is the sum of their variances after each has been scaled by the square of its coefficient in the sum. Specifically, if T = aX₁ + bX₂, where a and b are constants, then the variance of T, denoted as Var(T), is given by Var(T) = a²Var(X₁) + b²Var(X₂).

Given that X₁ has a mean of 10 and a standard deviation of 3, and X₂ has a mean of 12 and a standard deviation of 4, we apply the formula for the variance of T:

• Var(T) = (2)²(3)² + (3)²(4)²
• Var(T) = 4(9) + 9(16)
• Var(T) = 36 + 144
• Var(T) = 180

Therefore, the variance of T is 180, corresponding to option D.