ROUGH Not Delhi School of Ecomomics Course 003: Basic Econometrics Miktterm Exam

Question

ROUGH Not Delhi School of Ecomomics Course 003: Basic Econometrics Miktterm Exam: October 24, 2016 Maximiim Marks 16 Rohini Somanathan ns: Do all quaestions. They all carry equal marks. You bave 70 minates to complete the exam. UD be LLd. with a Unifocm distribution on the unit interval and X UU+...+ U Which important distribation approximates the distribution of X. Specify the distribution completely with its parameters. ive a simple but accurate approximation for P(x > 17). No elaborate calculations are necessary uppose the random variable X has a probability density fanction given by where the parameter >0. JrFind the maximum likelihood estimator of Does the maximum likelihood estimator for the median of this distribution exist? If yes, derive it, and if not, explain why % Suppose that Xi,..,x,form a random sample from anormal distribution for which the mean is unknown and the variance 2 is known. IIow large a random sample must be taken in order that there will be a confidence interval for with confidence coefficient 0.95 and length less than 001 , A single observation is taken from a uniform distribution on (0,0) and you would like to test the null hypothesis of against the alternative, 2. uf Find a test procedure for which Type error, a(6)-05. Characterize all the test procedures for which a(6-0, which one of these minimizes (6)? now that uhave a sampl ofsizen,the null hypothe sei of = 1 asuthe alterh uye hypotesis is composite with T-Graph thepower functionorthe test derivechpart (a)

Explanation / Answer

Ui ~ Unif(0,1)

E(Ui) = 1/2

Var(Ui)= 1/12

X = U1 + U2 +... U60

E(X) = 60*E(Ui) = 30

Var(X) = 60*Var(Ui) = 60*1/12 = 5

By central limit theorem

X ~ N(30,5)

b) P(X >17) = P(Z> (17- 30)/sqrt(5))

= P(Z> -5.81377674)

= 1        

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