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I've to show, if possible, that these two statements hold:

  1. Show that if $x \sim Uniform(0,1)$ is a data sample of size 10x10, further vectorized, then $$100 \le \int p_z(g)\sum_{i=1}^{100} (2x_i^2 + 1) dz \le 200.$$

  2. Show that if $z \sim \mathcal{N}(0,1)$ is a data sample of size 10x10, further vectorized, then $$100 \le \int p_{data}(x)\sum_{i=1}^{100} (z_i^2)dx \le 200. $$

The only think I know is that in both problems the summation can be taken out and the remaining integrals are equal to 1, thus $$ \int p_z(g)dz = \int p_{data}(x)dx = 1. $$

For problem 1, the lower bound can be easily shown replacing all $x_i$ with 0, but doing the same with 1 the upper bounds will come up to be 300.

For problem 2, I dunno how model boundaries for the normal distribution. Should I consider the expected value instead? Someone knows how to help me?

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