Although we can see the result from Brownian motion that small particles in the fluid actually have random movement, which means these fluctuations are unsymmetrical, I still don't know why in statistics we can say in small intervals of time these fluctuations would be unsymmetrical?
This happens because of the random motions of the molecules that hit the particles. That random motion is what we know as temperature. All molecules have a speed that is roughly the same, but the are going in random directions. When the size of an object gets small enough, then over short intervals, more molecules hit one side of the object than the other. The net result of all those collisions is not equal.
You can see this randomness in probability if you toss ten coins into the air. Do it 10 times. Each time, there will be some number of heads, and some number of tails. On average, the counts will trend toward 5 of each. But for any one throw, the value will range from 0 to 10 for each, with most values between 3 and 7.
So, if you drew a graph of the number of heads for each throw, it would be a squiggle. This is analogous to Brownian motion.
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