Suppose y is a random variable whose density is either:f(y | H₂) =[/3(y+1), 0≤x≤10,[1,f(y | H₂) =>= 0,else0≤y≤lelsepratenکلہاFind the Bayes rule and the resulting error probabilities Pr(Do | H₁) and Pr(D₁ | Ho) fortesting Ho versus H₁ with equal priors and uniform costs (Ci = 0, Cij = 1).

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Answered: Suppose y is a random variable whose…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let A E BL(H). Then A is called normal if A*A = AA*, uni- tary if A*A = I = AA*, that is, A* = A-¹, and self-adjoint if A* = A. Clearly, if A is unitary or self-adjoint, then A is normal. How- ever, a normal operator need be neither unitary nor self-adjoint. For example, let H = K² and for x = (x(1), x(2)) in H, A(x) = (x(1) − x(2), x(1) + x(2)). Then it can be easily seen that for x = H, A*(x) = (x(1) + x(2), −x(1) + x(2)), Request A*A(z) = 2(x(1),z(2)) = AA*(x). Hence A*A = AA*, but A*A# I and A* ‡ A. explain these steps in détail.
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Suppose that X is a discrete random variable with the follo wing probability mass fundion: 2 3 p(x) | 20/3 | 0/3 2 (1 - €) (3 T(1-8)/3 where OLeLI is a parameter. the following R independent observations were taken from such distribution: (3,0, 2,1,0,3,0,2,I, 0, 2, 1). Use the method of maximum likelihood to find the estimate of o.
X is a random variable such that X ~ B(n, p) where mean is 2 and 3 variance is . Find the probability of X is not less than 7. 2
Let X and Y be two random variables with joint probability mass function: p(x,y) = xy(1+y) for X=1,2,3 and Y=1,2 p(x,y) = 0, Otherwise. Please enter the answer to 2 decimal places. • What is the variance of (4-5X)?
Q3/ If the M.g.f. of random variable x is : M(t) = (0.75 + 0.25 e')" x = 0,1, 2,3, .,n 1- if (n = 4) find the Median and Pr(0 sx< 1)? 2-if (n = 5) find var(x), E X(X – 1) ? 3- find c .d. f. of X?
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b) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.

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