Concept explainers
The joint distribution for the length of life of two different types of components operating in a system was given in Exercise 5.18 by
The relative efficiency of the two types of components is measured by U = Y2/Y1. Find the probability density
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Chapter 6 Solutions
Mathematical Statistics with Applications
- Let X1, . . . , Xn be iid with pdf f(x) = 1 x √ 2πθ2 e − (log(x)−θ1) 2 2θ2 , −∞ < x < ∞, and unknown parameters θ1 and θ2. Find the maximum likelihood estimators for θ1 and θ2, respectivelyarrow_forwardA rectangular plate with insulated surface is 10 cm. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u(x,0) = 8 sin(px/ 10) when 0 <x <10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0o C, find the steady state temperature distribution u(x,y).arrow_forwardf X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + ···+ Xn is a sufficient estimator of the parameter θ.arrow_forward
- Consider a 6-meter metal bar with a uniform initial temperature (across the bar) of 35°C . Suppose it is in thermal contact with an external source of heat given by h(x)= 3−x, 0 ≤ x ≤ 6. So the temperature u(x,t) had the ut=uxx+h(x) permission. Suppose further that the temperature of the ends are kept constant, being at x=0 of 5°C , while at x=6 of 30°C . Under such conditions: Find the steady-state temperature distribution of the bar and the boundary value problem that determines the transient distribution. (no need to solve the problem).arrow_forwardLet yt = φyt−1 + et with et ∼ WN(0,σ2) and |φ| < 1. Consider the over-differenced process wt = (1 − L)yt.(i) What is the model followed by wt? (ii) Is wt invertible? (iii) Obtain V [wt] and compare its magnitude with V [yt] and hence comment on the impact of over-differencing on the variance of a stationary process.arrow_forwardSuppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. FInd (d) fZ|XY (z|x,y), (e) fX|YZ(x|y, z).arrow_forward
- Q / The probability of not matching the salts ratio to sand from a given source is 0.2 and the probability of its grain gradient not matching the specifications is 0.03 The probability of sand matching the specifications is equal to ?,arrow_forwarddW is normally distributed, dW has mean zero, dW has variance equal to dt. Parameter other than dw is assumed as constant. We have a representation of the geometric Brownian motion as dS/ S = µ dt + σ dW, prove µ dt + σ dW is normally distributed and find its mean and variance.arrow_forwardConsider X and Y are joint distributed with PDFf(x,y)=x+y, 0≤x≤1, 0≤y≤1. (a) Find the probability that X > 0.5. (b)FindP(X> Y 1/2).(c) Are X and Y independent?arrow_forward
- The critical value of t at a given alpha level will be smaller than the comparable critical value of z and is much smaller when n is very small.arrow_forwardFrom exercise 9-15 Find the boundary of the critical region if the type I error probability is a) alpha = 0.01 and n = 10 b) alpha = 0.05 and n=10 c) alpha = 0.01 and n=16 d) alpha = 0.05 and n=16 From exercise 9-15 standard deviation = 20 H0: mu = 175 H1: mu >175arrow_forward1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.arrow_forward
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