{x1#x2# ··· #xk | k ≥ 1, each xi ∈ {a, b}∗, and for some i and j, xi = x^R j }
1. Convert the CFG from the first part to a PDA. Give an example computation on a string of at least 9 symbols in the language of your PDA.
The random vector x (XI, X2,... ,Xk)' is said to have a symmetric multivariate normal distribution if x ~ Ne(μ, Σ) where μ 1k, i.e., the mean of each X, is equal to the same constant μ, and Σ is the equicorre- lation dispersion matrix, i.e. when k 3, μ-0, σ2-2 and ρ 1/2, find the probability that Hint: Recall that if x = (Xi, , Xk), has a continuous symmetric dis tribution, then all possible permutations of X1,... ,Xk...
For a given k, 1 ≤ k ≤ n, how many vectors (x1, x2, . . . , xk) are there for which each xi is a positive integer such that 1 ≤ x1 < x2 < · · · < xk ≤ n?
For a given k, 1 ≤ k ≤ n, how many vectors (x1, x2, . . . , xk) are there for which each xi is a positive integer such that 1 ≤ x1 < x2 < · · · < xk ≤ n?
7. Suppose that Xi,..., Xk are independent random variables, and X, ~ Exp(B) for i = 1, . . . , k. Let Y = min(X1 , . . . , Xk). Show that Y ~ Exp(Σ-1 β).
Let X1,X2,,,Xn be jointly noemal with EXi=0,EXi^2=1 for all i and cov(Xi,Xj)=ρ, i,j=1,2,,,(i≠j). What is the limiting distribution of n^-1Sn. Where Sn=∑Xk?
3. Suppose (X1, ..., Xk) follows a multinomial distribution with size n and event probability Pi, i = 1, ..., k. (C-1 Xi = n and Li-l pi = 1). (a) Show Xi~ Binom(pi) for i = 1, ..., k. (b) Show X; + X; ~ Binom(pi + pj), for 1 <i, j <k and i # j. (c) Show Cov(Xi, X;) = -npipj. (Hint: V(X; + X;) = V(X;) + V(X;) + 2Cov(Xi, X;)).
* SUBSET-SUM-kS, t> I S -[xi Xk] and for some lyı yn)cIxi.... xk) the sum of the yi's equals t. For example, <S-2, 3, 5, 7, 11, 14], t-21> is in SUBSET-SUM because 3+7 11-21. xk) can be partitioned into two parts A and -A where -A * SET-PARTITION <S> S-Ixi S-A and the sum of the elements in A is equal to the sum of the elements in A. For example, 〈 S-12, 3, 4, 7, 8/> works because...
Suppose (X1, ..., Xk) follows a multinomial distribution with size n and event probability pi, i = 1, ..., k. O , X; = n and k pi = 1) (a) Show X; ~ Binom(Pi) for i = 1, ..., k. (b) Show X1 + X; ~ Binom(Pi + Pj), for 1 Sinj 5 k and i øj. (c) Show Cov(Xi, X;) = -npipj. (Hint: V(X; + X;) = V(X;) + V(X;) + 2Cov(Xi, X;)).
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c) Sn=X1+X2 + . . . + Xn. (d) An -Sn/n
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c)...