Let $X_1,...,X_{20}$ be independent Poisson random variables with mean 1. Use the central limit theorem to approximate $$P\{\sum_{i=1}^{20} X_i \geq\ 15\}$$.
To solve this question, I need to find the expectation of $X = \sum_{i=1}^{20}X_i $, which I know is equal to 20, but I can't seem to figure out what the variance is.
In the solution manual, they use expectation and variance of $X_i$ in the formula for the central limit theorem, which they say are both equal to 1.
In the central limit theorem, do you use the expectation and variance of the $X_i$ or $X = \sum_{i=1}^{20}X_i $? Why is the variance of $X_i$ = 1 in this case? And also, what is the variance of $X = \sum_{i=1}^{20}X_i $?
For a Poisson random variable, the variance is always equal to the mean.
In general, the variance of the sum of independent variables is equal to the sum of the variances of all the variables.