# How to get better approximation than Central Limit Theorem

This is continuation of my problem

Calculate variance of sum random variables

Suppose random variable $$X$$ takes 3 values $$1, 2, 3$$ with probability $$\frac{1}{2}$$, $$\frac{1}{3}$$ and $$\frac{1}{6}$$. Sample 1000 times from $$X$$ independently and do the following operations. Take a number $$T=0$$. If 1 comes, add 0.18 with $$T$$, if 2 comes add 0.05 with $$T$$ if 3 comes add -0.29 with $$T$$. Then expected value of $$T$$ will be 58.33 and variance will be 27.65. Now I want to find the probability $$T<50$$. For this I tried to use central limit theorem namely I approximated the distribution by Normal $$N(58.33,27.65)$$. Then integrate the density function of this normal distribution from -infinity to 50 to get the probability. Can we use better approximation than this? Since we know higher moments of $$T$$, can we somehow use these values?

You are adding many i.i.d. random variables with nice distribution, so the CLT will work fantastically. If you are after an approximately exact result, you could have a look at the asymptotic expansions mentioned in the comments or FFT.

An other (simple) option is simulation: Here, we draw 100'000 times from the distribution of interest and calculate mean, variance and the probability to be below 50.

### R code

set.seed(0)
one_rv <- function(n = 1000) {
sum(sample(c(0.18, 0.05, -0.29), size = n,
replace = TRUE, prob = c(1/2, 1/3, 1/6)))
}

many_rv <- replicate(1e5, one_rv())
mean(many_rv)                             # Mean: 58.3512
var(many_rv)                              # Variance: 27.69896
mean(many_rv < 50)                        # Prob < 50 by simulation: 0.056600
pnorm(50, mean = 58.33, sd = sqrt(27.65)) # By normal approx:        0.056579


The probabilities are very close (and both versions are approximations).