mber of defects on the rear side (Y) of the panel are under study. 1. Suppose that the joint p.m.fof X and Y is modeled as fx (x, y) = s(x+ y), x = 1, 2, 3 and y = 1, 2, 3 Determine the value of c. 2. Determine the marginal p.mf of X. Find also the mean and variance of X. 3. Determine the conditional p.m.f. of Y given X= 2. Find also the conditional mean and conditional variance of Y given X= 2.
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- (a∗ ) Compute P(X = 4, Y = 5). (b∗ ) Compute the marginal pmf’s of X and Y . (c∗ ) Are X and Y independent? Let Z be the minimum of X and Y . Calculate E[Z].Determine S[y] has a local maximum or minimum on the stationary path. (by the value of the O(ϵ2) term in Δ)The number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are under study. Suppose that the joint pmf of X and Y is modeled as fxy (x,y)=c(x+y), x=1,2,3 and y=1,2,3.Determine the marginal pmf of X. Determine the conditional pmf of Y given X=2.
- In the article “Influence of Crack Width on Shear Behaviour of SIFCON” (C. Fritz and H. Reinhardt, High Performance Fiber Reinforced Cement Composites: Proceedings of the International RILEM/ACI Workshop, 1992), the maximum shear stress τ of a cracked concrete member is given to be τ = τ0(1 − kw), where τ0 is the maximum shear stress for a crack width of zero, w is the crack width in mm, and k is a constant estimated from experimental data. Assume k = 0.29 ± 0.05 mm−1. Given that τ0 = 50 MPa and w = 1.0 mm, both with negligible uncertainty, estimate τ and find the uncertainty in the estimate.As an industrial engineer is working a test of material used to on a certain product to determine whether it is more economical to repair a damaged area rather than replace the entire structure. The engineer obtained the following data on the bond length (x variable) vs. the rupture load (y variable) Bond Length (mm) Rupture Load (N) 50 17,052 50 14,063 80 26,264 80 19,600 110 21,952 110 26,362 140 26,362 140 26,754 170 31,654 170 32,928 Find the 95% Confidence Interval for the conditional mean of the response variable at x = 140A number of studies have shown lichens (certain plants composed of an alga and a fungus) to be excellent bioindicators of air pollution. The article “The Epiphytic Lichen Hypogymnia physodes as a Biomonitor of Atmospheric Nitrogen and Sulphur Deposition in Norway” (Environ. Monitoring Assessment, 1993: 27–47) gives the following data (read from a graph) on x ¼ NO3 wet deposition (g N/m2 ) and y ¼ lichen N (% dry weight): (refer to chart) The author used simple linear regression to analyze the data. Use the accompanying MINITAB output to answer the following questions: a. What are the least squares estimates of b0 and b1? b. Predict lichen N for an NO3 deposition value of .5. c. What is the estimate of s? d. What is the value of total variation, and how much of it can be explained by the model relationship?
- Find the critical point of ƒ(x, y) = xy + 2x - ln x2y in the open first quadrant (x >0, y>0) and show that ƒ takes on a minimum there.Use the method of variation of parameters to find a particular solution of the DE…Approximate the critical points of g(x) = x cos−1 x and estimate the maximum value of g.
- Determine the yy''' + 2x = tanx dependent variable independent variable type pde/ode order degree linearityUse Improved Euler's Method to approximate a solution y(x) to the problem of initial value [image] at points x0 = 2, x1 = 2.5 and x2 = 3.For the production function Q (K,L)= 1000 (KL3 )1/3 , Given the current input level K = 400 and L = 40. Use marginal analysis to estimate the approximate change in output which results form an increase in K of 9 units and a decrease in L of 7 units. Verify that Young′s Theorem holds?