[Solved] MA331 Homework 02

Problem 1. Assume that N B(n,p), a Binomial distribution with number of trials n and probability of success p. Set p = 0.4.

• For n = 20,30,50,75,100, accurately compute P(N 8.25) by using R function.
• For n = 20,30,50,75,100, approximate P(N 8.25) by using Laplace theorem.
• Evaluate and scatter plot errors of all approximations of (ii), i.e., the absolute difference between the accurate computation and the Laplace approximation.
• What do you perceive based on the scatter plot of errors in (iii).

Problem 2. Check the instruction of R commands plot(density(x)). Generate a SRS of size n for the population X N(2,3²), and evaluate the samples of

, , ,

respectively. Then, based on the corresponding samples, plot estimated density curves of and , respectively, and also make the scatter plot of .

• For n = 20, simulate and for 100 times.
• For n = 30, simulate and for 100 times.
• For n = 50, simulate and for 100 times. (iv) For n = 75, simulate and for 100 times.
• Based on the plots of and in (i) (iv) describe your findings on probability distributions of and , respectively.
• Based on the scatter plots of in (i) (iv) describe your findings on the statistical association between and .

Problem 3. Show that E[N] = np for N B(n,p).

Problem 4. Show that E[T] = 0 for T T_n.