Concept explainers
When X1, X2,…, Xn are independent Poisson variables, each with parameter μ. and n is large, the sample mean
has approximately a standard normal distribution. For testing H0: μ = μ0, we can replace μ by μ0 in the equation for Z to obtain a test statistic. This statistic is actually preferred to the large-sample statistic with denominator
Want to see the full answer?
Check out a sample textbook solutionChapter 8 Solutions
Probability and Statistics for Engineering and the Sciences
- Assume that ξ has a rectangular distribution with the mean value E(ξ) = 0 and the variance V(ξ) = 12. Calculate P (ξ> 5).arrow_forwardIf X is a random variable with pdf f(x) = 2x − 2 where x = (1, 2), find the variance of Y = 2X - 3.arrow_forwardShow that the mean x of a sample of size n from the distribution. e-x/0 f (x)=- A i 0arrow_forward8. Let y be a normal random variable with a constant mean E(y) = 4, constant variance var(y.) = o², and a covariance cov(y, y;) = 0 for t# j. Consider the sample mean j = E %3D A. Show that the variance of the sample mean y is .arrow_forwardThe random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−3/4. Calculate Var(X+Y) with a full explanationarrow_forwardLet X be a random variable with mean E[X] = 20 and variance Var(X) = 3. Define Y = 3 – 6X. Calculate the mean and variance of Y.arrow_forwardAmericans consume an average of 1.64 cups per day. Assume the variable is normally distributed with a standard deviation of 0.24 cup. If 500 individuals are selected, how many will drink less than 1 cup? Let X and Y be two random variables having joint pdf f(x, y) = (3x-y)/9 1arrow_forwardA simple random sample of 100 water meters within a community is monitored to estimate the average daily water consumption per household over a specified dry spell. The sample mean and sample variance are found to be ȳ=12.5 and s2=1252. If we assume that there are N=10,000 households within the community, estimate μ, and place bound on the error of estimation.arrow_forwardSuppose that X1,..., X, is a random sample from a normal distribution with mean u and variance o?. Two unbiased estimators of o? are ởf = s° = E(x, - Xy°, and ô = (X1 - 2 Find the efficiency of ở relative to ôž.arrow_forwardLet X₁, X2,..., Xñ be a random sample from the exponential distribution with rate parameter A. (a) Find the CRLB for the variance of all unbiased estimators of X. (b) Find the efficiency of the MLE of X. (c) Find the CRLB for the variance of all unbiased estimators of E(X) = 1/λ. (d) Can you find an estimator of E(X) = 1/λ that is efficient? Justify your answer.arrow_forwardThe variable X has a normal distribution with mean u = 100 and standard deviation o = 8. The variable Y has a normal distribution with mean u = 85 and standard deviation o = 6. Variables X and Y are independent. (a) Find the mean, u, and standard deviation, o, of the sum X + Y. = = (b) Find the mean, u, and standard deviation, a, of the difference X - Y. O =arrow_forwardIf X is a Gaussian random variable with mean zero and variance oʻ, find the pdf of Y = \X\.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill
Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill