6) Given: (a) f (x) = (2x^2)/(x^2 −1) - Calculate f ′(x) and f
″(x) - Determine any symmetry - Find the x- and y-intercepts - Use
lim f (x) x→−∞ and lim f (x) x→+∞ to determine the end behavior -
Locate any vertical asymptotes - Locate any horizontal asymptotes -
Find all intervals where f (x) is increasing and decreasing - Find
the open intervals where f (x) is concave up or concave down
consider the function f(x) = 1 +
x3 e-.3x
a. what is f'(x)
b. what is f''(x)
c. what are the critical points of f(x)
d. are the critical points a local min or local max or
neither?
e. find the inflection points
f. if we define f(x) to have the domain of [2,50] compute the
global extreme of f(x) on that interval
A joint pdf is defined as f(x) =cxy for x in [1,2] and y in
[4,5]
(a) What is the value of the constant c?
(b) Are X and Y independent? Explain.
(c) What is the covariance oc X and Y? i.e. Cov(X ,Y)
Let X1, ..., Xn be iid with pdf f(x; θ) = (1/ x√ 2π)
e(-(logx- theta)^2) /2 Ix>0 for θ ∈ R.
(a) (15 points) Find the MLE of θ.
(b) (10 points) If we are testing H0 : θ = 0 vs Ha :
θ != 0. Provide a formula for the likelihood ratio test statistic
λ(X).
(c) (5 points) Denote the MLE as ˆθ. Show that λ(X) is can be
written as a decreasing function of | ˆθ|...
Expand the function, f(x) = x, defined over the interval 0 <x
<2, in terms of:
A Fourier sine series, using an odd extension of f(x)
and A Fourier cosine series, using an even extension of f(x)
The joint PDF of X and Y is given by f(x, y) = C, (0<
x<y<1).
a) Determine the value of C
b) Determine the marginal distribution of X and compute E(X) and
Var(X)
c) Determine the marginal distribution of Y and compute E(Y) and
Var(Y)
d) Compute the correlation coefficient between X and Y
Let X and Y have the joint PDF
f(x) = { 1/2 0 < x + y < 2, x > 0, y > 0 ;
{ 0 elsewhere
a) sketch the support of X and Y
b) Are X and Y independent? Explain.
c) Find P(x<1 and y<1.5)