need help with foundation of statistics assignment due at midnight. includes Rstudio. Fairly textbook problems. Please let me know.
Question 1 (20 pts)
Random variable X follows a normal distribution: X = N(2,42), that is, µ = 2,‡ = 4
(a) (10 pts) Calculate the probability P(|X| Æ 3). Note: be careful, it is |X|. Use two methods to work on it below.
Method 1 (Theory): Using the transformation of general normal distribution to standard normal distribution method, finally use R code to calcuate it.
Method 2 (Simulation): Using simulation method (rnorm function) to estimate. Write down your R code and final result.
(b) (5 pts) Find the value k such that P(X k) = 0.2 (using the transformation of standard normal distribution method), and finally use R code to calculate it.
(c) (5 pts) Use simulation method to estimate the mean: E(e?X2).
Question 2 (20 pts)
Random variable T describes the life time of a laptop, which follows Exponential distribution. That is, T = Exp(—), where — = 6 (years).
(a) (10 pts) If a company bought 90 laptops of same type, find the probability that at least 43 laptops are still working at the end of 9 years? Also, what is the mean number of laptops that are still working at the end of 9 years in this company?
(b) (10 pts) Use simulation method to estimate the probability P(3 T 10) and the mean E(T3).
Write down your R code and final results.
Question 3 (20 pts)
Let X be a random variable with a density function (pdf)
f(x) = ce?x/2,x 0 (a) Calculate the value of c to make f(x) a valid pdf.
(b) Calculate the probability P(X Æ 6|X 3)
Question 4 (40 pts)
(a, 20 pts) In an exam, there are 100 questions of same di culty levels, and assuming the time to answer these questions are independent random variables. Also, the time to answer each questions follow the same distribution with the same mean µ = 35 seconds, and standard deviation ‡ = 15 seconds. Question: what is the probability that you can answer all the 100 questions within 1 hour? Finally you should use R code of standard normal distribution to calculate the final value.
(b, 20 pts) In a year, the mean SAT Math score for the female students is 518, standard deviation is 15; while the mean SAT Math score for the male students is 525, standard deviation is 25. If we randomly draw 100 females and 100 males from all the students who took the test. Calculate the probability that the male students’ average score are at least 10 points higher than female students’ average score. Finally you should use R code of standard normal distribution to calculate the final value.