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An Introduction to Mathematical Statistics and Its Applications (6th Edition)
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- If X is a uniformly distributed random varibale with a=8 and b=13, then Calculate the mean and variance of X? Round to three decimal placesarrow_forwardIf X is a uniformly distributed random varibale with a=9 and b=16, then Calculate the variance of X? Round to three decimal placesarrow_forwarda hypothesis test produces a t statistic of t=2.3. if the researcher is using a two tailed test with a=0.05 how large does the sample have to bw in order to reject the null hypothesis?arrow_forward
- Let X be an exponential random variable with standard deviation σ. FindP(|X − E(X)| > kσ ) for k = 2, 3, 4, and compare the results to the boundsfrom Chebyshev’s inequality.arrow_forwardLet X1, X2, … , Xn be uniformly distributed on the interval 0 to a. Recall that the maximum likelihood estimator of a is a X ˆ = max i ( ).(a) Argue intuitively why aˆ cannot be an unbiased estimator for a.(b) Suppose that E a na / n () ( ) ˆ = +1 . Is it reasonable that aˆ consistently underestimates a? Show that the bias in the estimator approaches zero as n gets large.(c) Propose an unbiased estimator for a.(d) Let Y X = max( )i . Use the fact that Y y ≤ if and only if each X y i ≤ to derive the cumulative distribution function of Y . Then show that the probability density function of Y isf ynya , ya,nn ( ) = ≤ ≤⎧⎨⎪⎩⎪−100 otherwise Use this result to show that the maximum likelihood estimator for a is biased.(e) We have two unbiased estimators for a: the moment estimator a X ˆ1 = 2 and an n X ˆ2 =+ 1 i [( ) / ] ( ) max , where max( ) Xi is the largest observation in a random sample of size n. It can be shown that V() ( ) a a/ n ˆ1 2 = 3 and that V ( ) [ ( )] a a / nn ˆ2 2 = + 2 .…arrow_forwardA hypothesis test produces a t statistic of t=2.01.If the researcher is using a two tailed test with x=0.05,how large does the sample have to be in order to reject the null hypothesis?arrow_forward
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