How to Solve Continuous Random Variables — Univariate RV — gamma (Erlang) tail probability Questions on Exam P
Sample Practice Problem
ID: #HVZ3G
Let \(X\) and \(Y\) be the two independent backup-pump exponential operating times, each with mean 4 hours. Calculate the variance of their total operating time.
📖 Worked Solution & Strategy
An exponential random variable with mean \(\mu = 4\) has variance \(\operatorname{Var}(X) = \mu^2 = 16\). For independent random variables \(X\) and \(Y\), the variance of a linear combination is \(\operatorname{Var}(aX + bY) = a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)\). Here \(\operatorname{Var}(X+Y) = 1^2(16) + 1^2(16) = 32\) and \(\operatorname{Var}(X-Y) = 1^2(16) + (-1)^2(16) = 32\). The variance of the total operating time \(X+Y\) is \(\operatorname{Var}(X+Y) = 32\). The evaluated result is \(32\), which matches the correct option.