statistics and probability questions using R studio

18. Every year, hundreds of thousands of graduating college seniors ponder over one of the most intriguing questions - what is a good GRE score? The Educational Testing Service gives a general [GRE test percentage distribution](https://www.ets.org/s/gre/pdf/gre_guide_table4.pdf) of scores by intended graduate field. For the Social and Behavioral Sciences the mean score for Verbal Reasoning section ($X_v$) was $153$ with a standard deviation of $8$ and the mean score for the Quantitative Reasoning section ($X_q$) was $152$ with a standard deviation of $9$. Suppose that both distributions are nearly normal and the performance on these two sections is independent. The questions below are for students who intend to study the Social and Behavioral Sciences.

a) Assuming you choose to model these scores with normal distributions, write down the distributions of $X_v$ and $X_q$, specifying all the parameters.

b) What percent of of individuals have a Quantitative Reasoning score greater than $162$?

c) What Verbal Reasoning score should one have to be in the top $10\%$?

d) The Combined score is the sum of the Verbal Reasoning and Quantitative Reasoning score (that is, $C=X_v + X_q$). What is the mean of the Combined score (that is, $E(C)$)?

e) Using the assumption that performance on these two sections is independent, explain why the variance of the Combined score is $145$.