04) a) Prove the following relations N, a i)Σ a. =11-as, otherwise 1-a b) Find the...
Is the following series cos n convergent or divergent? Prove your result. 2 if Σ an with an > o is convergent, then is Σ a.. always convergent? Either prove it or give a counter example. 3 Is the following series convergent or divergent? if it is divergent, prove your result; if it is convergent, estimate the sum. 4 Is the following series 2n3 +2 nal convergent or divergent? Prove your result.
Let Σ = {0, 1). (a) Give a recursive definition of Σ., the set of strings from the alphabet Σ. (b) Prove that for every n E N there are 2" strings of length n in '. (c) Give a recursive definition of I(s), the length of a string s E Σ For a bitstring s, let O(s) and I(s) be number of zeroes and ones, respectively, that occur in s. So for example if s = 01001, then 0(s)...
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
11. (6 points) Find the sum of the following series: (a) Σ 2n +1 3η n=0 ΟΙ (5) Σ n! ΠΟ
,X, ,n. independent, the central Xi, E(X)=0, var(X)-σ are Prove 3. Assume <o。 13<oo, 1=1, limit theorem (CLT) based EX1 result regarding what are conditions on σ that we need to assume in order for the x.B.= Σσ, as n →oo. In this context, X,, B" =y as n →oo, In this context, result to hold?
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(ii) Use (i) (otherwise no credit will be given!) to show that cos(20) = 2 cos - 1. 3. Prove that if z + is real, then z is real or |z| = 1. 4. Identify all the point in the complex plane which satisfy the following relations: (i) z2 = 2(2-1), 12-11 212 +11, (iii) z +11 <4 - 12 - 11.
1. Complete the following exercises a) For Σ = {a, b} find regular expressions for the compliment of the following languages L = L(aa*bb) b) Let Li = L(ab*aa), L2 = L(a"bba"). Find a regular expression for (L1 n Ljl2. c) The symmetric difference of two sets Sı and S2 is defined as sı Θ s,-(x : x E Si or x E S2 but x is not in both S1 and S2). Show that the family of regular languages...
Let Bn be the number of equivalence relations on the set n. Prove that Bn = Bn-k k-1
Let Bn be the number of equivalence relations on the set n. Prove that Bn = Bn-k k-1
Denoting the variance of by ơ, prove that n' ) σ ơy _ (N-1) n State (without proof) the expected value of the sample variance s2. Derive an unbiased estimator, so, for σ,.
Denoting the variance of by ơ, prove that n' ) σ ơy _ (N-1) n State (without proof) the expected value of the sample variance s2. Derive an unbiased estimator, so, for σ,.
Question 5 /10 points/ MLE of Variance is Biased For each n-1.M, let Xn ~ N(μ, Σ) denote an instance drawn (independently) from a Gaussian distribution with mean μ and convariance Σ. Recall /IML Xm, and Show that EML]-NN Σ Y ou mav want to prove, then use . where àn,m = 1 if m n and = 0 otherwise.
Question 5 /10 points/ MLE of Variance is Biased For each n-1.M, let Xn ~ N(μ, Σ) denote an instance...