Assume X1, . . . , Xn iid normal with mean and variance ^2 , show that
a. X¯ and X^2 are independent.
b. Proof that X¯ is normally distributed with mean and variance ^2/n.
c. Proof that (n ? 1)S^2/?2 is chi-squared distributed with (n ? 1) degrees of freedom.
d. Show that X¯ S/is t distributed with (n ? 1) degrees of freedom
This transformation is called Helmert's transformation (orthogonal transformation).
Under this orthogonal transformation,
Assume X1, . . . , Xn iid normal with mean and variance ^2 , show...
Problem 1. Assume, the observations X1, X2, . . . , Xn are iid. normal distributed random variables with unknown mean θ. You observe n = 16 many variables with the empirical mean 1.45 and a sample variance of 0.512. a) Determine a 90% two-sided confidence interval for the mean. b)HowcanwedecideonthehypothesisH0 :μ=2vsH1 :μ̸=2onthe significance level 10%, using just the answer for part a) and no additional computations? c) Now assume that, instead of using the sample variance, you know that...
X1,...,Xn are IID with N(0,2). a) Determine the mean and variance for (X (subscript 1)^2) b) Show sqrt(n) * [ log ( 1/n ∑(from i=1 to n) Xi2) − log(σ2 ) ] d → N(0, 2). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
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andom variables Xi, .. . , Xn are iid with mean fu- 12 and variance ?2 = 5. Let X (X + Xn)/n denote the average and T- Xi+.. + Xn denote the total of the variables. For parts (a) through (d) below, calculate the required probability and state whether your calculation is exact or approximate (a) P(X> 13) for n 5 if the underlying distribution is normal (b) P(T < 300) for n -20 if the underlying distribution is...
Suppose that Xi are IID normal random variables with mean 2 and variance 1, for i = 1, 2, ..., n. (a) Calculate P(X1 < 2.6), i.e., the probability that the first value collected is less than 2.6. (b) Suppose we collect a sample of size 2, X1 and X2. What is the probability that their sample mean is greater than 3? (c) Again, suppose we collect two samples (n=2), X1 and X2. What is the probability that their sum...