Let P be the plane in R3 whose coordinates satisfy the equation x + 3y -2z = 5, find a parametric representation for P. Find the parametric representation for a line in P which passes through the point [7 0 1].
Let p = 47 and q = 59, n = p . q, and e = 157.
a. Compute a multiplicative inverse of d, modulo φ(n).
b. Every two-letter string (including A-Z and spaces) can be converted to a number-message between 0 and 2626, by replacing a space by 00, ‘A’ by 01, ‘B’ by 02, etc.. For example. ‘ME’ becomes 1305. Encrypt the two-letter string ‘HI’ by computing its number-message m, and the ciphertext me mod n.
c. Decrypt the sequence of ciphertexts 0802, 2179, 2276, 1024 to find a message.
3.10. Let’s assume that you have been asked to calculate risk-based capital ratios for a bank with the following accounts:Cash _ $5 millionGovernment securities _ $7 millionMortgage loans _ $30 millionOther loans _ $50 millionFixed assets _ $10 millionIntangible assets _ $4 millionLoan-loss reserves _ $5 millionOwners’ equity _ $5 millionTrust-preferred securities _ $3 millionCash assets and government securities are not considered risky. Loans secured by real estate have a 50 percent weighting factor. All other loans have a 100 percent weighting factor in terms of riskiness.a. Calculate the equity capital ratio.b. Calculate the Tier 1 Ratio using risk-adjusted assets.c. Calculate the Total Capital (Tier 1 plus Tier 2) Ratio using risk-adjusted assets.
SOLUTION:Equity Capital Ratio = Equity Capital / Total AssetsEquity capital Ratio = ($5) / ($5 + $7 + $30 + $50 + $10 + $4)Equity Capital Ratio 4.72% Tier 1 Ratio = Tier 1 Capital / Risk…
Let S and T be subsets of the universal set U. Draw an appropriate Venn diagram and use the given data to determine the number of elements in each basic region.n(S) = 9, n(S ∪ T) = 28, n(T) = 22, n(S ‘ ∪ T ‘) = 30.
n( S ∩ T ) = n( S ) + n(T ) − n( S ∪ T )⇒ n( S ∩ T ) = 9 + 22 − 28 = 3
Prove that the ring R is not Noetherian. Prove that the ring R is not Noetherian. Prove that the ring R is not Noetherian. Prove that the ring R is not Noetherian.
5 . Let RE Q` be the subring consisting of polynomials } = dot a12t . .. + and`"such that do EI . Given a subgroup A in O ( viewed as an abelian group viaaddition ) , consider the subset I CR consisting of polynomials ajax t … + and`such that aj EA.
Let random variables X and Y respectively denote the number of heads and tails obtained in a sequence of n independent flips of a fair coin.
(a) What kind of discrete random variable is X? Specify its range, PMF values,mean and standard deviation.
(b) Let D = X−Y denote the difference between the number of heads and tails. Usingthe central limit theorem, find a suitable approximation for the probabilitythat |D| > m, where m is a positive integer m.
(c) Evaluate this probability for the following special situation involving a very large number of flips: n = 1020 and m = 109(i.e. the difference between the number of heads and tails exceeds one billion!)
Binomial approximation using central limit theorem1 Subject : Probability and StatisticsAssignment Title: Binomial approximation using central limit theoremCompilation Date: 17Th December 2015…
1. Let’s examine the market for books. Below are the demand and supply functions in the market for books. Qd=20-pQs=1/3pa. Find the market equilibrium price and quantity. Find consumer surplus, producer surplus, and total welfare at the equilibrium. Is there deadweight loss in this market (in other words, is this the efficient quantity)? b. Suppose the government wants to promote reading and so it subsidizes the suppliers of books. Find the new equilibrium quantity with an $4 per unit subsidy. What is the price consumer’s pay and the price that producer’s receive? c. What are the new consumer surplus, producer surplus, and total welfare? Is there deadweight loss in this market? If so, how much?
1. Let’s examine the market for books. Below are the demand and supply functionsin the market for books.Qd=20-pQs=1/3pa. Find the market equilibrium price and quantity. Find consumer…
let R be the region bounded by the graphs of y = sin(pie times x) and y = x^3 – 4x.a) find the area of Rb) the horizontal line y = -2 splits the region R into parts. write but do not evaluate an integral expression for the area of the part of R that is below this horizontal line.c) The region R is the base of a solid. For this solid, each cross section perpendicular to x-axis is a square. Find the volume of this solid. d) the region R models the surface of a small pond. At all points in R at a distance x from the y-axis, the depth of the water is given by h(x)=3-x. find the volume of the water in the pound.
Let Q be a symmetric transition probability matrix, that is qy qu for all i,j- 1,… ,N. Consider a Markov chain which, when the present state is i, generates the value of a random variable X such that P(X j) dy, and if X-j, then either moves to state j with probability b,/(b, + bi), or remains in state i otherwise, where b,, j-1,…, N, are specified positive numbers. Show that the resulting Markov chain is time reversible with stationary probabilities a Cb,, j1,.., N
2. Let Q be a symmetric transition probability matrix, that is, 53,- }- = q},- for alli, j. Consider a Markov chain which, when the present state is i, generatesthe value of a raindom variable X such that P{X = j} = (3,}, and if X = j,then either moves to state j with probability b}- /(b,- + bf), or remains in state1′ otherwise, where b}, j = l . . . , N, are specified positive numbers. Show that the resulting Markov chain is time reversible with limiting probabilities
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