Unformatted text preview: Mathematical Statistics Quiz #2 5/22/2008 6:30‐9:30PM 1. (a) State the definition of a random sample and a statistic. (b) State the Central Limit Theorem (CLT). (c) Prove the CLT by moment generating function. (d) Give the definitions of χ , t v , F , . How do you obtain these random variables with distributions from the normal distribution(s). 2. Let X and Y be independent continuous random variables with marginal density functions and . (1) (i) If X and Y are positive random variables, let U X Y, show that U has pdf by DF method. (ii) Let X and Y be two independent exponential random variables with parameter λ, find the distribution of U X Y using (1) (i). Identify the distribution. (2) (i) Let Z
Y X . Show that Z has the pdf || . (ii) Suppose that X and Y are independent standard normal random variables. Find the pdf of Z
Y X using (2) (i). Identify the distribution. 3. If X and S are the sample mean and sample variance of a random sample of size n from a normal population with mean and variance. (1) Show that X and S are independent. (2) What are the distributions of X and S ? Prove them. 4. (a) Find the moment generating function of the following distributions: (i) Poisson (ii) Normal (iii) Gamma(α, λ with pdf ,X , 0, 0. (b) Let X , be a random sample from exponential distribution with pdf , show that the distribution of 2λ ∑ X is . 5. If a random variable of is and . Show that (1) If (2) If is the time to failure of a product and the pdf and the CDF , respectively, then its failure rate at time t is given by has an exponential distribution, the failure rate is constant. has a Weibull distribution with pdf 0, the failure rate is given by . , 0, 0, 6. Let X , X , X be iid with pdf , 0, zero elsewhere. (1) Find the joint pdf of the random variables Y , Y , Y defined by Y
X X X X ,Y X X X X and Y X X X . Are they mutually independent? (2) Find the joint pdf of the random variables Y , Y , Y defined by Y
X X X ,Y X X X X X and Y X X X . Are they mutually independent? ...
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