Letπi , i = 1, . . . , n be positive numbers that sum to 1. LetQbe an irreducible

transition probability matrix with transition probabilities q(i, j ), i, j =

1, . . . , n. Suppose that we simulate a Markov chain in the following manner:

if the current state of this chain is i , then we generate a random variable that

is equal to k with probability q(i, k), k = 1, . . . , n. If the generated value is j

then the next state of the Markov chain is either i or j , being equal to j with

probability π j q( j,i )

πi q(i, j )+π j q( j,i ) and to i with probability 1 − π j q( j,i )

πi q(i, j )+π j q( j,i ) .

(a) Give the transition probabilities of the Markov chain we are simulating.

(b) Show that {π1, . . . , πn} are the stationary probabilities of this chain.

3. Let 31;, i = l, . . . , n be positive numbers that sum to 1. Let Q be an irreducibletransition probability matrix with transition probabilities q(i, j ), i, j =l, . . . , 11. Suppose that we simulate a Markov chain in the following manner:if the current state of this chain is i, then we generate a random variable thatis equal to k with probability q(i, k), k = 1, . . . , at. If the generated value is jthen the next state of the Markov chain is either i or j, being equal to j with probability W W) 31′ jfiIUJ) —:r;q(i,j)+njq(j,i) and to; With probab111ty l — —Kt_q(1.7j) +Jrthjai)’ (a) Give the transition probabilities of the Markov chain we are simulating.(b) Show that {31], . . . , 31”} are the stationary probabilities of this chain.

 

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Let A be the cross-sectional area of your U-tube and L be the length of the portion of the tube that is filled

with fluid. (The total volume of fluid in the tube is thus V= LA .) Show that if the fluid levels are displaced

from equilibrium by “±z ” (see Fig. 1), the increased weight of fluid on one side of the tube compared to the

other is: F=−ρ(hA)g=−2ρAgz,

(where ρ is the density of the fluid and “h ” is the instantaneous height difference between the fluid levels. This

excess weight on one side of the system is what causes the entire volume of liquid in the U-tube to oscillate!)

 

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Let A be an arbitrary m x n matrix, and let B be an arbitrary n x p matrix.

a. Show that any vector x that is in the column space of AB is also in the column space of A.

b. Is it also true that any vector x that is in the column space of AB is also in the column space ofB? Why or why not?

c. Is it also true that any vector x that is in the column space of A is also in the column space ofAB? Why or why not?

 

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Let ABC be a right triangle with ∠B = 90◦. Let E and F be respectively the mid-points of AB and AC. Suppose the incentre I of triangle ABC lieson the circumcircle of triangle AEF. Find the ratio BC/AB.6. Find all real numbers a such that 3

 

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let a1 = 2, a(n+1) = (5*a(n) – 4) / a(n) for n>=1. Show that (1 <= a(n) <= a(n+1) <= 4) for n>=1

Given,A(n+1) = 5*A(n) -4/A(n) A(n+1) = 5 – [4/A(n)]That can be written as[A(n+1) – 1]/4 = 1 – [1/A(n)]Putting A(1) = 2A(2)&gt; 1From A(2) ,A( 3) &gt;1This tells that A(n) &gt; 1A(n+1)…

 

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