Total time for a three-leg trip

Question

A trip has three independent legs, each with a normally distributed duration (in minutes): $N(20, 4)$ , $N(15, 1)$ , and $N(25, 9)$ , where the second number in each is the variance. Let $T$ be the total travel time.

What are the mean and variance of $T$ , and what is its distribution?

Accepted Answer

A sum of independent normals is normal. Add the means and add the variances: \mathbb{E}[T] = 20 + 15 + 25 = 60, \qquad \operatorname{Var}(T) = 4 + 1 + 9 = 14. So T \sim N(60,\ 14), standard deviation \sqrt{14} \approx 3.74. The rule extends to any number of independent pieces: means and variances each just accumulate term by term. Takeaway. For three independent normal legs, the total is normal with mean 60 and variance 14; sum the means and sum the variances.

Total time for a three-leg trip

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