Question

Find the variance of the random variable \(\mathrm{X}+\mathrm{b}\) where \(\mathrm{X}\) has variance, \(\operatorname{Var} \mathrm{X}\) and \(\mathrm{b}\) is a constant.

Short answer

The variance of the random variable X + b, where X has variance Var(X) and b is a constant, is equal to the variance of the random variable X, which is \(\operatorname{Var}(X)\). This is because the variance of a constant is always 0, and the variance of a sum of independent random variables and/or constants is the sum of their variances.

Step-by-step solution

Identify the random variables and the given variance

The random variable X has a given variance, denoted by Var(X), and the constant b has no given variance value since it's a constant.

Find the variance of the constant b

As mentioned earlier, the variance of a constant value is always 0. So, \(\operatorname{Var}(b)=0\).

Apply the variance properties

The variance of a sum of independent random variables and/or constants is the sum of their variances: \(\operatorname{Var}(X+b) = \operatorname{Var}(X) + \operatorname{Var}(b)\).

Calculate Var(X+b)

Now, we add the variances we found in Steps 1 and 2: \(\operatorname{Var}(X+b) = \operatorname{Var}(X) + 0 = \operatorname{Var}(X)\). Therefore, the variance of the random variable X + b is equal to the variance of the random variable X, which is \(\operatorname{Var}(X)\).