The standardized Gaussian PRNG produces Gi and the PRNG value $Y _ { i }$ is uniformly distributed over [0,1).If $\mathrm { G } _ { i } \text { and } Y _ { i }$ are independent, what is the mean and variance of $Z _ { i } = 2 G _ { i } + 4 Y _ { i } - 1$

Question

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The Answer is

2 \mathrm { G } _ { i }

has mean 0 and variance

ext { 4. } 4 \mathrm { Y } _ { i }

has mean 4(0.5)=2 and variance

16 ^ { * } ( 1 / 12 ) = 4 / 3 ext {. Adding } - 1

gives

\mu _ { z } = 1 ext { and } \sigma _ { z } ^ { 2 } = 4 + 4 / 3 = 16 / 3

The Standardized Gaussian PRNG Produces Gi and the PRNG Value YiY _ { i }Yi​

The Standardized Gaussian PRNG Produces Gi and the PRNG Value $Y _ { i }$