Correct, it makes no difference. This indifference is part of what makes a hash function a "cryptographic hash function". The distribution is nearly gaussian, close enough to call it random.
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you probably wanted to say uniform distribution.
Hash function have the design goal of inducing a probability distribution on their image which is closer to a uniform distribution as possible.
So the probability that h(X) = k is the same for every k.
continuous uniform distribution
which is commonly associated with gaussian normal distributions.
of course it's discrete integers but continuous implies total randomness since all bits are at play and not any holes in it or symmetries.
Probability distribution - Wikipedia