Download Homework 1 Solutions - Topics in Biostatistics | STATS 449 and more Study notes Statistics in PDF only on Docsity! Homework 1 STATS 449 (Due 1-19) Instructions: You can work together on homework, but your final answers must be your own. Late homework cannot be accepted. The Courseinfo.pdf document on our CTools site gives guidelines about the format of homework assignments. Make sure you show R code and output where relevant. 1) Let X be an exponential random variable with parameter λ = 1/10. That is, X ∼ Exp(1/10). a) State the values of µ = E[X] and σ2 = V ar(X). No need to perform any calculations. Use information given in Lecture 2 notes. Now suppose X1, X2, . . . , Xn are independent and identically distributed (iid) exponential random variables with parameter λ = 1/10. Let X̄ = 1 n n∑ i=1 Xi b) State the values of E[X̄] and V ar(X̄) according to the information given in class, or by using Slide 32 in Lecture 2 notes. V ar(X̄) should depend on n. c) Now, set n = 100 and demonstrate the central limit theorem. Using R, generate 100,000 independent Exp(1/10) random variables and put them in a 100× 1000 matrix. Calculate X̄ for each column, giving a sample of size 1,000 from the distribution of X̄. Calculate the mean and standard deviation of the sample. Make a histogram of the sample drawn from the distribution of X̄. Explain why the results support the central limit theorem. Turn in the output from R showing the mean, the standard deviation, and the histogram. Hint: See the R code in Lecture 2. To generate exponential random variables use x <- rexp(100000,rate=0.1) 2) The Poisson distribution is a good approximation to the binomial when n is large, p is small, and the Poisson parameter λ is set equal to np. a) Suppose that the probability that an item produced by a certain machine will be defective is 0.03. Find the probability that a sample of 30 items will contain less than or equal to 1 defective item. b) Repeat part a) using the Poisson approximation. Is the approximation close?
