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An Introduction to Mathematical Statistics and Its Applications (6th Edition)
- Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?arrow_forwardIf X is a random variable having the standard normaldistribution and Y = X2, show that cov(X, Y) = 0 eventhough X and Y are evidently not independent.arrow_forwardSuppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variablesarrow_forward
- Let Mx, y be the moment generating function of random variables that are not independent of X and Y. Which of the following / which are not the properties of the function Mx, y?arrow_forwardData collected over a number of years show that whena broker called a random sample of eight of her clients,she got a busy signal 6.5, 10.6, 8.1, 4.1, 9.3, 11.5, 7.3, and5.7 percent of the time. Assuming that these figures canbe looked upon as a random sample from a continuous uniform population, use the estimators obtained in Exer-cise 57 to estimate the parameters α and β.arrow_forwardA single observation of a random variable havinga geometric distribution is to be used to test the null hypothesis that its parameter equals θ0 against the alter-native that it equals θ1 > θ0. Use the Neyman–Pearson lemma to find the best critical region of size α.arrow_forward
- Assume that the random variables X1 and X2 are bivariate normally distributed with mean µ and covariance matrix Σ. Show that if cov(X1, X2) = 0 then X1 and X2 are independent.arrow_forwardSuppose that the makers of Duracell batteries want to demonstrate that their size AAbattery lasts an average of at least 45 minutes longer than Duracell’s main competitor,the Energizer. Two independent random samples of 100 batteries of each kind areselected, and the batteries are run continuously until they are no longer operational. Thesample average life for Duracell is found to be 308 minutes. The result for the Energizerbatteries is 254 minutes. Assume the entire AA batteries standard deviation 84 minutesand Duracell batteries 67 minutes. Is there evidence to substantiate Duracell’s claim thatits batteries last, on average, at least 45 minutes longer than Energizer batteries of thesame size? (assuming equal population variances)arrow_forwardSuppose Y is a continuous random variable drawn from the uniform distributionon the interval [3, 4], that is, Y ∼ Uniform([3, 4]). Conditioned on Y = y, a second randomvariable X is drawn from the uniform distribution on the interval [0, y]. What is fX(x), thepdf of X?arrow_forward
- Describe a bivariate random variable that has cdf H(x,y) = max(0,x) ×max(0,y) forx < 1 and y < 1, and 1 elsewherearrow_forwardA random group of apartments was selected from a city to analyze the number of bedrooms they have. there evidence to reject the hypothesis that the apartments are equally distributed between 1-bedroom, 2 bedroom and 3-bedroom apartments , at alpha = 0.05arrow_forwardIf X1 and X2 are independent random variables hav-ing binomial distributions with the respective parameters n1 and θ and n2 and θ, show that Y = X1 + X2 has thebinomial distribution with the parameters n1 + n2 and θ.arrow_forward
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