First, computing an estimate of a stochastic variable is not the same as estimating the distirbution of the variable.  You ETA is an estimate of a stochastic it's just one number, aka a statistic, preferably something like the expected value, but not necessarily. Waze looks at the speed of cars along each segment of your future route and calculates an average duration for traversing each segment.  It adds these to the present time and that's your ETA. There's not necessity that these durations be statistically independent.  They add up to an estimate of the expected value anyway. Brent On 7/10/2018 8:54 PM, Henry Baker wrote:

I find myself driving quite a bit on freeways, and I'm curious about how my car GPS can calculate a reasonable ETA (Estimated Time of Arrival) for me.

I learned calculus in high school & college, but I don't recall learning about how to integrate a *probabilistic* variable -- i.e., a variable which is subject to some probability distribution.

So here's the thing: suppose I'm driving at a speed whose values come from some distribution, and I want to integrate this speed over some amount of time to compute another probability distribution about how much distance is being covered within that time, or to estimate a distribution of my arrival time at a particular destination. (Let's assume that time itself moves precisely, but the car speed varies.)

Now an integral is merely a fancy *sum*, and we know how to estimate the sum of *independent* random variables.

But in the case of my car, the speed at one moment is NOT independent of the speed at the next moment! The acceleration, maybe, but not the speed. More likely, even the acceleration isn't independent from moment to moment, but the "jerk" (dacceleration/dt) might very well be.

So now we have an even bigger problem: we need to go back an integrate the jerk(t) for each moment of time t.

But wait, there's more!

If I'm using a program like Waze, which has a probability distribution for the speed on each segment of each highway, then I should be somehow computing an overall distribution for my trip time from some sort of integration of all of these little contributions.

If these individual distributions were independent, then presumably my result would be some sort of convolution of all of the little distributions.

But the distributions aren't independent.

(Even worse: Waze knows that the speed distributions are time-dependent -- e.g., based upon time of day. But I don't even know how to do the simpler problem.)

So what now?

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