We are given two uncorrelated random variables, $\epsilon$ and $\lambda$, with variances $Var[\epsilon] = 2$ and $Var[\lambda] = 3$. We want to find the variance of the linear combination $3\epsilon + \lambda$.

1. Problem Description

We are given two uncorrelated random variables, $\epsilon$ and $\lambda$, with variances $Var[\epsilon] = 2$ and $Var[\lambda] = 3$. We want to find the variance of the linear combination $3\epsilon + \lambda$.

2. Solution Steps

Since $\epsilon$ and $\lambda$ are uncorrelated random variables, the variance of their sum is the sum of their variances.

Also, the variance of a constant times a random variable is the square of the constant times the variance of the random variable.

The formula for the variance of a linear combination of uncorrelated random variables is:

$Var[aX + bY] = a^2 Var[X] + b^2 Var[Y]$, where $X$ and $Y$ are uncorrelated.

In our case, we have $Var[3\epsilon + \lambda]$.

Let $a = 3$, $X = \epsilon$, $b = 1$, and $Y = \lambda$.

$Var[3\epsilon + \lambda] = 3^2 Var[\epsilon] + 1^2 Var[\lambda]$

$Var[3\epsilon + \lambda] = 9 Var[\epsilon] + Var[\lambda]$

We are given that $Var[\epsilon] = 2$ and $Var[\lambda] = 3$.

Substituting these values, we get:

$Var[3\epsilon + \lambda] = 9(2) + 3 = 18 + 3 = 21$.

3. Final Answer

The variance of $3\epsilon + \lambda$ is

2

1.

Final Answer: The final answer is $\boxed{21}$