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Unformatted text preview: 550.430 Introduction to Statistics Spring 2012 Naiman Homework #4 Due Monday Feb 27th (1) The following question was raised in Lecture #6. If X is a random variable, is it always the case that Cov( X p ,X q ) ≥ 0 if p and q are positive integers? (a) Consider the case when X ∼ N ( μ,σ 2 ) . Write down E [ X ] , E [ X 2 ] and E [ X 3 ] in terms of μ and σ. Simplify the covariance between X and X 2 and explain why this can be negative in some cases. (b) Suppose X is a nonnegative random variable. If ψ 1 and ψ 2 are monotone increas- ing functions, show that Cov( ψ 1 ( X ) ,ψ 2 ( X )) ≥ . Hint: Let Y be independent of X and have the same distribution as X. Explain why the random variable Z = ( ψ 1 ( X ) − ψ 1 ( Y ))( ψ 2 ( X ) − ψ 2 ( Y )) is always nonnegative. Conclude that E [ Z ] ≥ . Next, simplify this expectation to get a conclusion about the desired covariance. (c) Use (b) to show that if X is a nonnegative random variable, then Cov( X p ,X q ) ≥ if p and q are positive integers. (2) A physics experiment is carried out several times under two conditions. Under the first condition, particle counts are obtained and can be assumed to be Poisson distributed with unknown parameter λ > . Under the second condition, the counts are also assumed to be Poisson distributed, but the parameter becomes cλ where c > 0 is also unknown. The experiment is carried outunknown....
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This note was uploaded on 02/26/2012 for the course STATISTICS 530 taught by Professor Naiman during the Fall '11 term at Johns Hopkins.
- Fall '11