Question: Given a pair of random variables X and Y having
Given a pair of random variables X and Y having the variances σ21 and σ22 and the correlation coefficient ρ, use Theorem 4.14 to express var(X/s1 + Y/s2) and var(X/s1 – Y/s2) in terms of σ1, σ2, and ρ. Then, making use of the fact that variances cannot be negative, show that –1 ≤ ρ ≤ + 1.
Answer to relevant QuestionsGiven the random variables X1, X2, and X3 having the joint density f(x1, x2, x3), show that if the regression of X3 on X1 and X2 is linear and written as Then Where µi = E(Xi), σ2i = var(Xi), and σij = cov(Xi,Xj). ...Making use of the fact that = y – β and β Sxy/Sxx , show that Use the results of Exercises 14.20 and 14.21 and the fact that E(Bˆ) = β and var(Bˆ) = σ2/ Sxx to show that Yˆ0 = Aˆ + Bˆ is a random variable having a normal distribution with the mean And the variance Also, use the ...By solving the double inequality –zα/2 ≤ z ≤ zα/2 (with z given by the formula on page 402) for ρ, derive a (1 – α) 100% confidence interval formula for ρ. On page Various doses of a poisonous substance were given to groups of 25 mice and the following results were observed: (a) Find the equation of the least squares line fit to these data. (b) Estimate the number of deaths in a group ...
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