What is the expected value and standard deviation of a weighted binomial distribution?

Question

I'm working on a genetics problem and am using binary representation of the genes i.e. they can be 0 or 1. With 20 genes, I have an expected additive genetic value (i.e. mean) of 10 with standard deviation of 2.236 and variance 5 because this is a binomial distribution.

However, the genes don't all contribute equally so I'm weighting them from a negative exponential distribution with lambda = 0.405. In R code this is:

x <- c(1:20)
lambda <- 0.405
expo <- function(x,lambda) {
  y<-exp(-lambda*x)
  print(y)
}

The sum is then 'gene1 * 0.67 + gene2 * 0.44 + ... + gene20 * 0.0003' where the genes can be 1s or 0s.

My question is how do I calculate the summary statistics for this new distribution of weighted values?

Answer 1

Note that we have$$\mathbb{E}\left( \sum_{i=1}^n c_i X_i\right)= \sum_{i=1}^n c_i \mathbb{E}(X_i)$$

$$Var\left( \sum_{i=1}^n c_i X_i\right)= \sum_{i=1}^n\sum_{j=1}^n c_ic_j Cov(X_i, X_j)$$

Hence, you should look for covariance information between the genes to compute your desired quantity.