Concept explainers
Let X and Y have a bivariate
Find
(a)
(b)
(c)
(d)
Want to see the full answer?
Check out a sample textbook solutionChapter 4 Solutions
Probability And Statistical Inference (10th Edition)
- For the following table of data. x 1 2 3 4 5 6 7 8 9 10 y 0 0.5 1 2 2.5 3 3 4 4.5 5 a. draw a scatterplot. b. calculate the correlation coefficient. c. calculate the least squares line and graph it on the scatterplot. d. predict the y value when x is 11.arrow_forward2. Let X be a random variable with pdf fx(x), and Y = X². %3D (a) Find fx(x|X > 0) (b) Find fy(y|X > 0)arrow_forwardGiven the two h-scatterplots below, would you say the data is spatially autocorrelated in the direction chosen? Why? h 10 m h 20 m 10 10 024 10 20 NParrow_forward
- The variance of Y is σ2y = 4.6875 and the variance of the sum is σ2x + y = 9.1875. You knowthe variance of X=1. Find the covariance, xy. Show workarrow_forwardSuppose that X and Y have the following joint probability distribution: f(x.y) X 2 4 1 0.10 0.15 2 0.20 0.30 3 0.10 0.15 Find the marginal distribution of X and Y. Find the expected value of g(x.y) = xy? or find E(xy²). Find µx and µy. Find Cov(x,y) Find the correlations p(x,y)arrow_forward- (Sec. 6.1) Using a long rod that has length µ (unknown), you are going to lay out a square plot in which the length of each side is µ. Thus the area of the plot will be µ². However, because you do not know the value of µ, you decide to make n independent measurements X1,...,X, of the length. Assume that each X; has mean µ and variance o². (a) Show that X² is not an unbiased estimator for the area of the square plot µ². [Hint: for any rv Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.] (b) For what value of k is the estimator X² – kS² unbiased for µ²?arrow_forward
- If the PDF of X is f(x)=2x/k2 for 0<x<k, for what value of k is the variance of X equal to 2?arrow_forwardNow, consider an estimator of μ: W=1/16Y1+1/16Y2+1/4Y3+1/8Y4+1/2Y5 This is an example of a weighted average of the Yi’s. Show that W is also an unbiased estimator of μ. Find the variance of W.arrow_forwardlet x has the p.d.f (3xe-v3x 0 < x < 0 0. W f(x) = By m.g.f find the mean and variance of the function?arrow_forward
- The moment generating function can be usedto find the mean and variance of the normal distribution.a. Use derivatives of MX(t) to verify that E(X)=meanand V(X) =varianceb. Repeat (a) using RX(t)=ln[MX(t)], and comparewith part (a) in terms of effort.arrow_forwardConsider a random sample from NB(r, p) where the parameter r is known to be 3. An experiment is run with n = 10 trials and the sample mean is observed to be x̄ = 0.6. (a) Derive a formula for the MLE p̂ as a function of n, r and X̄ . (b) Find the estimate of p. (c) Find the MLE for the population mean.arrow_forwardIf X is a Poisson variable such that P(X =2) = 9P(X= 4) + 90P(X = 6), find the mean and variance of X.arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill