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comment How to determine the step response using convolution of the signal's impulse response?
@Rajesh Properly answering requires more space, but simply put: tao is your integration variable, the thing you change as you perform the infinite summation, and t is a "constant", because the whole integral is to calculate the response at a specific point in time (t). Thankfully, it is "parametrized", so you can evaluate the integral for any t of your choosing (meaning you can calculate the value of the response at any instant).
comment Spread evenly $x$ black balls among a total of $2^n$ balls
For now let me define dispersion as follows: If 0 means move to the left a distance of d0 and 1 means move to the right d1, and d0 and d1 are chosen so that after moving 2^n-x times to the left and x times to the right you end up in the same position, minimal dispersion is the same as minimizing maximum excursion from origin over time, if the pattern is repeated endlessly.