Cauchy. 4. (4 points) Suppose X ~ (a) Find P(X E [0, 8]). (Hint: tan-"(a) =...
7. (5 points) Find the linear approximation for f(x) = tan(2x) at a = 0 and use it to approximate the value of tan(0.002). Hint: The linear approximation is just the tangent line to the curve at a = 2. 8. (5 points) Use the Mean Value Theorem for derivatives to find the value of x = c for f(x) = Vx on the interval (1,9). 9. (5 points) The acceleration of an object moving along the number line at...
(Calculus 1)
tan(3x) 1 - 8-21 e-2 Find the derivative of the integral function H (x) = 4+e-21 dt sin(3x)
Show that, if an ≥ 0 for all n ∈ N and (an) is a Cauchy
sequence, then (√ an) is also a Cauchy sequence. Hint: x − y = (√ x
− √y)(√ x + √y)
Show that, if an > 0 for all n є N and (an) is a Cauchy sequence, then (Van) is also a Cauchy sequence. Hint: r -y- (V1-vu) (Va + vⓙ
Show that, if an > 0 for all n є N and...
Problem III. (12 points) Consider the following probability distribution. X 0 6 P(X = x) 1/4 1/4 1/4 1/4 1. (2 points) Find E(X). 2. (5 points) Find the sampling distribution of the sample mean X for samples of size = 2. n = 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1, ... , Xn) is proposed as @(X1, ... , Xn) = -XI(X; #0, and X: #6), п i=1 where...
The standard Cauchy distribution has cumulative
distribution function
F(x) = 1 + 1 tan−1(x) 2π
where −∞ < x < ∞.
1
a. Find the probability density function of X.
b. FindxsuchthatP(X>x)=0.2.
Question 7: The standard Cauchy distribution has cumulative distribution function F(x) = + tan-1 (x) π where-00< 00. We were unable to transcribe this image
(Hint: think of tan as a fraction!) lim (1 - .) tan 7. Suppose a function f(x) satisfies f'(x) < 0 and f(x) > 0 for all x 20. > 0 for all x > 0. Consider the function F(x) := ["s(t) dt whose domain is 0,00). Is F(x) an increasing or decreasing function
Use the graph below to find 8>0 such that for all x, 0< x-cl<6Ifx)- LI<e The value of & is (Simplify your answer) y 3x-4 20- 12.2 11H 9.8 fix) 3x-4 c-5 HI L-11 III c-12 4 5 6 4.6 5.4 2 8 10
Use the graph below to find 8>0 such that for all x, 0
4. (*) Solve the Cauchy problem Ut = 3Uxx, X E R, t> 0, u(x,0) = Q(x), x E R, for the following initial conditions and write the solutions in terms of the erf function. LS 2, -4 < x < 5 (a) $(x) = { 0, otherwise. (b) (x) = e-la-11 Note: In (b) complete the square with respect to y in the exponent of e to obtain a nice form. You need to split your integral based on...
Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion defined by bo - 1 and bn- k-0 n E N. Show that bn-- Hint: Use a) with e*e*1 and the inverse of a power series found in the lecture.
Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion...
This is a MATLAB question so please answer them with MATLAB
steps.
Let f(z) = V3z sin(#) and P(z) =r-x-1. 1. Find f(e) 2. Find the real solution(s) to Px) 0. Hint: use the roots command. 3. Find the global minimum for f(x). Hint: plot f over [0,2] 4. Solve f()P. Hint: plot f and P over [0,21. 5. Find lim,→0+ f(x). Hint: make a vector hi make a table [x + h; f(x + h)]". 6. Find '(In 2)....