Using R or Rstudio to compute Mathematical Statistic Questions.

timer Asked: Nov 11th, 2016

Question description

Using R or Rstudio to compute some probability questions. The questions are in the pdf file.

The ID number for the set.seed is: 00479185

After finish the work on R or Rstudio, save the file and I will need to print it out.

Math 448 • Mathematical Statistic Computing Homework 2. Due Friday Nov 11 1. Let X1 , X2 , · · · be independent Bernoulli trials with p = 0.5, and let Yn denote the number of 1’s in the first n trials: Yn = X1 + X2 + · · · + Xn . Let p̂n = Yn /n. (a) Precede your R script with set.seed(your BU student ID #). (b) Generate 1 sample. What is the observed value of p̂1 ? (c) Repeat the process 5 times, each time updating your p̂ (in other words, generate 5 more Bernoulli variates, and calculate p̂ based on your past observations). You will end up with values for X1 , · · · , X6 : one from the first part, and five more from this one. What value of X7 , if any, will result in p̂7 > p̂6 ? Generate X7 ; what value did you observe? (d) Sequentially generate X’s up to n = 100 samples in total, and compute p̂ after every sample (overall, you will need to generate 100 variates in this problem). Plot p̂, and comment on consistency Yn /n. 2. Let X1 , X2 , · · · be independent standard normal random variables. Let A2n denote the sample variance, Bn2 denote the MLE of the variance, and Cn2 = (X1 − X2 )2 /2. (a) Precede your R script with set.seed(your BU student ID #). (b) Generate 2 samples. What are the observed values of A22 , B22 , C22 ? Calculate the difference between A22 and B22 . (c) Generate one more sample. What are the values of A23 , B32 , C32 ?. (d) Sequentially generate n = 1000 samples in total, updating A2 and B 2 with each observation. Plot: (i) A2n ; (ii) Bn2 ; and (iii) their difference. Make conclusions.

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