A box contains 5 r balls 3 w balls and 2 b balls calculate


Part A -

1. Assume a matrix T (that transforms probability densities) is given by

528_figure.png

(a) Calculate the determinant, Δ, of the matrix T, i.e., Δ = det T.

(b) Calculate the eigenvalues, λ1 and λ2, of T as a function of Δ.

(c) Assume the eigenvectors v1 and v2, corresponding the eigenvalues λ1 and λ2 respectively, are written as

297_figure1.png

calculate the parameters a and b.

(d) Normalize v1 and v2 such that their "lengths" are 1, i.e., v112 + v212 = 1 + a2 = 1 for v1 while vie v122 + v222 = b2 +1 = 1 for v2.

(e) Normalize v1 and v2 such that their "sums" are 1, i.e., 1 + a = 1 for v1 while b + 1 = 1 for v2.

(f) Write the matrix Λ (for the eigenvalues) as

790_figure2.png

and the matrix V (for the eigenvectors) as

1386_figure3.png

and show that the two eigen equations can be written as

T V =V Λ,

(g) Assuming the vectors are normalized according to (e), calculate the inverse of V, i.e., V-1.

Part B -

1. A box contains 5 R balls, 3 W balls, and 2 B balls. The event Xi represents the ball designated in the process of selecting a ball from the box, noting the ball, and returning the ball to the box. Thus, the set Xxi = {R, W, B), with probabilities Pxi = (p, q, r} = {0.5, 0.3, 0.2). An iid process consist of a sequence X1, X2, ... X8.

(a) Calculate the entropy

H3 (p, q) = -p log p - q log q -(1 - p - q) log(1 - p - q),

(b) Calculate the range of probabilities of the joint density p(X1, X2, . . .X8).

(c) Calculate the range of joint densities for the members of the typical set A(8), for ∈ = 0.1, 0.5, and 1.

(d) Calculate the mass density of the typical set for the three values of ∈.

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Engineering Mathematics: A box contains 5 r balls 3 w balls and 2 b balls calculate
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