Suppose I have some quantity $Z$ I'm estimating that has an uncertainty $\sigma_Z = f(X)$, where the function $f(X)$ is known. Say that $X$ also has some uncertainty $\sigma_X$ (which has some numerical value). Can I calculate a numerical value for $\sigma_Z$?
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$f(X)$ is known or $f$ is known? – shimao Sep 11 '20 at 04:27
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The function $f$ is known (so both). Sorry I'll clarify in the question. – wht Sep 11 '20 at 04:30
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It will depend on both the function $f$ and distribution of $X$. For some simulation, I have $f(X) = X^2$, and $X \sim \text{Poisson}(4)$, or $X \sim \mathcal{N}(0, 4)$. For both $X$, the standard deviation is 2. However, as numerically shown, standard deviation of $f(X)$ depends on the specific distribution $X$.
set.seed(1)
f <- function(x){
x ^ 2
}
num_sim <- 10000
x_pois <- rpois(num_sim, 4)
z_pois <- f(x_pois)
print(paste0('sd(x_pois): ', sd(x_pois), ', sd(z_pois): ', sd(z_pois)))
x_norm <- rnorm(num_sim, mean=0, sd=2)
z_norm <- f(x_norm)
print(paste0('sd(x_norm): ', sd(x_norm), ', sd(z_norm): ', sd(z_norm)))
[1] "sd(x_pois): 2.01302632310105, sd(z_pois): 18.822808849991"
[1] "sd(x_norm): 1.97799168067352, sd(z_norm): 5.50156007079308"
If you know the specific parametric form of distribution of $Z$, you should be able to derive the distribution of $f(Z)$, and calculate the standard deviation. Otherwise, it would be difficult.
david
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