Similarly, if we look at the probability with which the parody, of the bits of x,
that is the XOR, of all the bits in x, is 1.
Well, if D is a pseudorandom distribution then that probability should
approximate the probability with which,
the parody of x is 1, when x is sampled from the uniform distribution.
That probability of exactly one-half.
So therefore if D is pseudorandom, the probability with which the parody of
x is 1 when x is sampled from D, should also be close to one-half.
And more generally we can fix, some set, of statistical tests.
That is, any kind of predicate, we define on a string x.
And we can look at or we can compare, the probability with which Ai of x is equal to
1 when x is sampled from distribution D and the probability that A ek,
Ai of x is equal to 1, when x is sampled from the uniform distribution.
And those should be close.
And if we require that to hold, if we require closeness to,
to hold for say 20 different statistical tests A1 through A20,
then that can serve as our definition ,of what it means for D to be pseudorandom.
That is D is pseudorandom, if these probabilities are equal for
i equals 1 to 20 or are close rather, for i equals 1 to 20.