STA 4322 Exam I Name: Introduction to Statistics Theory

Transcription

1 STA 4322 Exam I Name: Introduction to Statistics Theory Fall 2013 UF-ID: Instructions: There are 100 total points. You must show your work to receive credit. Read each part of each question carefully. There are 5 problems on 7 pages. The last page is Table 4 from the textbook. You are responsible for checking that your exam is complete. The following abbreviations are used throughout: CLT = central limit theorem iid = independent and identically distributed mgf = moment generating function pdf = probability density function N = {1, 2, 3,... } You may use the following facts/formulas without proof: Gamma density: X Gamma(α, β) means X has pdf 1 f(x α, β) = Γ(α) β α xα 1 e x/β I (0, ) (x) where α > 0 and β > 0. Also, E(X) = αβ and Var(X) = αβ 2. The Square of a Standard Normal: If Z N(0, 1), then Z 2 χ 2 (1). t Distribution Let Z and W be independent random variables such that Z N(0, 1) and W χ 2 (ν). Then T = Z W ν has a t distribution with ν degrees of freedom. We write this as T t(ν). F Distribution Let W 1 and W 2 be independent random variables such that W 1 χ 2 (ν 1 ) and W 2 χ 2 (ν 2 ). Then F = W 1/ν 1 W 2 /ν 2 has an F distribution with ν 1 numerator degrees of freedom, and ν 2 denominator degrees of freedom. We write this as F F (ν 1, ν 2 ). Covariance: If U and V are random variables, then the covariance of U and V is defined as Cov(U, V ) = E [( U E(U) )( V E(V ) )]. Variance of a sum: If X 1..., X n are random variables, then ( n ) n n 1 n Var X i = Var(X i ) + 2 Cov(X i, X j ). i=1 i=1 i=1 j=i+1 Moments of a random variable: Let X be a random variable. For p N, the quantity EX p is called the pth moment of X.

2 STA 4322 Exam I 2 1. Let X and Y be continuous random variables with joint pdf given by { e y 0 < x < y < f(x, y) = 0 otherwise (a) Neatly sketch a graph showing the support of (X, Y ). Clearly label the axes as well as any functions that you draw. (Recall that the support is the set of possibilities for (X, Y ); that is, the set of values in R 2 that (X, Y ) can take.) (b) Find the marginal pdfs of X and Y. (c) Identify the marginal pdfs of X and Y as members of a parametric family of densities that we learned about in class. (d) Find the covariance of X and Y. (e) Define W = X + Y. In class, we learned about a useful two number summary of a probability distribution. Provide that summary for the probability distribution of W. (35 pts)

3 STA 4322 Exam I 3 2. Recall from calculus the definition of the hyperbolic cosine function: cosh(t) = et + e t Carefully describe an experiment and a corresponding random variable, X, having mgf M X (t) = cosh(t). (10 pts) 2.

4 STA 4322 Exam I 4 3. We learned in class that if X 1, X 2,..., X n are random variables, and a 1, a 2,..., a n are constants, then E ( n i=1 a ) ix i = n i=1 a ie(x i ). You may use this result without proof throughout this question. Let Y be a random variable. (a) Write down the definition of the variance of Y. (b) What is the smallest variance that a random variable can possibly have. Provide an example of such a random variable by giving its probability distribution. (c) Show that the variance of Y can be written as the second moment of Y minus the square of the first moment of Y. (d) Let c and d be constants. Show that the variance of cy + d is equal to c 2 times the variance of Y. (25 pts)

5 STA 4322 Exam I 5 4. Let T and F be random variables such that T t(ν) and F F (1, ν). Prove or disprove the following statement: T 2 and F share the same probability distribution. (10 pts)

6 STA 4322 Exam I 6 5. ( HW Problem 7.57) Twenty-five heat lamps are connected in a greenhouse so that when one lamp fails, another takes over immediately. (Only one lamp is turned on at any time.) The lamps operate independently, and each has a mean life of 50 hours and standard deviation of 4 hours. Use the CLT to approximate the probability that there will still be a lamp burning 1300 hours after the lamp system is started. (You do not need a calculator - only simple arithmetic.) (20 pts)