A probability density function on R is a function f :R -> R satisfying (i) f(x)≥0 or all x e R and (ii) \int_(-\infty )^(\infty ) f(x)dx = 1. For which value(s) of k e R is the function

f(x)= e^(-x^(2))\root(3)(k^(5)) a probability density function? Explain.

Let X = ( X1, X2, X3, ,,,, Xn ) is iid,

f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b < 1

then, find a two dimensional sufficient statistic for (a, b)

Suppose that you pick a bit string from the set of all bit strings of length ten. Find the probability that

1. the bit string has exactly two 1s;
2. the bit string begins and ends with 0;
3. the bit string has the sum of its digits equal to seven;
4. the bit string has more 0s than 1s;
5. the bit string has exactly two 1s, given that the string begins with a 1.

Given a complicated lim x → af (x), how do you quickly determine which items are

significant and which items are negligible —¿ quick guess in 5 seconds? Degree (or order) of largeness. How to convert your QuickGuess to an air tight high quality argument?

Problem 3Consider the following definitions for sets of characters:•Digits ={0,1,2,3,4,5,6,7,8,9}•Letters ={a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}•Special characters ={∗,&,$,#}Compute the number of passwords that satisfy the given constraints

.(i) Strings of length 7. Characters can be special characters, digits, or letters ,with no repeated characters

.(ii) Strings of length 6. Characters can be special characters, digits, or letters ,with no repeated characters. The first character can not be a special char-acter.

Determine whether the following two planes x + 4y − z = 7 and 5x − 3y −7z = 11 are parallel, orthogonal, coincident (that is, the same) or none of these.

please show full working for learning purposes

Use the technique developed in this section to solve the minimization problem.

The minimum is C =

at (x, y, z) = .

find the general solution of the given differential equation

1. 2y''+3y'+y=t^2 +3sint

find the solution of the given initial value problem

1. y''−2y'−3y=3te^2t, y(0) =1, y'(0) =0

2.  y''−2y'+y=te^t +4, y(0) =1, y'(0) =1

Let S1 be the level surface given by the equation f(x,y,z) =k1and letS2be the level surface given by the equation g(x,y,z) =k2. Suppose that C is the curve of intersection of the surfacesS1andS2andassume that P is a point on C. show that any tangent vector to C at P must be parallel to∇f(P)×∇g(P)

Solve each of the following linear systems by the method of the example on page 283.

(a) 3x − y = 4        x + 5y = 9

(b) x − 2y + z = 1       y + z = 0         −x + y + z =−3

HInt; first find the QR decomposition of the coefficient matrix via the Gram–Schmidt process

Use ten iterations of the appropriate MATLAB function, with x^(0)=[0,...,0]', to solve Ax=b (approximately).

A)use Jacobi iteration.

B) use Gauss-siedel iteration.

1) make sure to use SOR with w=1.25, w=1.5, w=1.75,w=1.9, and optimal value if given.

* A=[1,-2,0,0;-2,5,-1,0;0,-1,2,-0.5;0,0,-0.5,1.25]] , B=[-3;5;2;3.5]. , (optimal w is 1.5431.)

A) Use Jacobi or Gauss-Seidel iteration and perform three iterations by hand.

B) Use Jacobi or Gauss-Siedel iteration for ten iterations with a MAT-LAB function.

* A= [5, -1,0;-1,5,-1;0,-1,5] , B=[9;4;-6]

Allow the main process to generate a text file containing the text of this assignment.

The main process will then create two other processes and pass to them the name of the file and two codes (code1 and code2) using a pipe() system call. The codes are two different alphabetic letters such as “a” and “k”.

• The child processes should search the file for the character in the interval received, compute their frequencies and return them through a separate pipe to the parent process.

• The parent process should compute the total number of characters in the file, and the rate of the appearance of the character frequencies received from the two child process, through separate pipes.

• The parent process should then form a table of alphabetic characters and their frequencies and print the table in a proper format on the screen.

Hint: You are expected to use fork(), pipe(), read(), write(), close(), waitpid(), and other appropriate system call, if needed.