How does the Central limit theorem work when there are multiple random variables, each one with a different mean and variance?
For example, let $X_1, X_2$ and $X_3$ be random variables that represent the income of three shops per day. Each $X_i$ is independent of each other and are normally distributed with $\mu_1 = \$1500000, \mu_2=\$2500000, \mu_3 =\$2000000, \sigma_1=\$300000,\sigma_2=\$150000$ and $\sigma_3=\$200000$. What is the probability that the income of the three shops during 5 days is greater than $\$29720000$?
My thoughts: $\sigma_\bar{X}^2=\sum\limits_{i=1}^3\sigma^2_{X_i}=1.525\times10^{11}$ and $\mu_\bar{X}=\sum\limits_{i=1}^3\mu_{X_i}=6000000$
Normalizing, I have $\frac{29720000-6000000}{390512.484}=60.74$
What am I doing wrong?
Thanks