[SOLVED] MA331 Homework 02

Description

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.

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