# Normal distribution of multiple events

I'm currently working on this problem and am having a bit of difficulty getting started. I found the normal distributions for both Mike and Jack with a score of over 225, but I'm unsure how to proceed with them. How should I be taking the normal distribution keeping in mind both of their scores will total to over 225?

Mike's bowling scores are normally distributed with mean 110 and standard deviation 13, while Jack's scores are normally distributed with mean 135 and standard deviation 10. If Mike and Jack each bowl one game, find the probability that the total of their scores is above 225 (assuming their scores are independent)

Any direction would be great! Thanks

## 1 Answer

Hint: If $X$ and $Y$ are independent, normally distributed, with means $\mu_X$, $\mu_Y$ and variances $\sigma_X^2$, $\sigma_Y^2$ respectively, then $X+Y$ has normal distribution, mean $\mu_X+\mu_Y$, and variance $\sigma_X^2+\sigma_Y^2$.

More generally, if $a$ and $b$ are constants, then $aX+bY$ is normally distributed, with mean $a\mu_X+b\mu_Y$, and variance $a^2\sigma_X^2+b^2\sigma_Y^2$.