Question

Provide an example of a function, y = f(x), that is defifined on [5,8], f(5) < 0, f(8) > 0 and

f is never zero. Explain why your example does not violate IVT.

Answer #1

Let f(x) be a continuous, everywhere differentiable function.
What kind information does f'(x) provide regarding f(x)?
Let f(x) be a continuous, everywhere differentiable function.
What kind information does f''(x) provide regarding f(x)?
Let f(x) be a continuous, everywhere differentiable function.
What kind information does f''(x) provide regarding f'(x)?
Let h(x) be a continuous function such that h(a) = m and h'(a) =
0. Is there enough evidence to conclude the point (a, m) must be a
maximum or a minimum?...

f (x, y) =(x+y)/(5(5 + 2)) is a joint probability density
function over the range 0 < x < 5 and 0 < y < 2. Find V
(X). Please report your answer to 3 decimal places.

A joint density function of the continuous random variables
x and y is a function f(x,
y) satisfying the following properties.
f(x, y) ≥ 0 for all (x, y)
∞
−∞
∞
f(x, y) dA = 1
−∞
P[(x, y) R] =
R
f(x, y) dA
Show that the function is a joint density function and find the
required probability.
f(x, y) =
1
8
,
0 ≤ x ≤ 1, 1 ≤ y ≤ 9
0,
elsewhere
P(0 ≤...

Give an example or explain why the request is impossible.
A function f(x) that is continuous at 0 and g(x) that is not
continuous at 0 such that f(x) - g(x) is continuous at 0.

Show an example of F(x,y) defined on [0,+∞)× [0, +∞) such
that
(i) F(0,0) = 0,
(ii) F(+∞,+∞)=1
(iii) for every x ≥ 0, F (x, y) is increasing in y; for every y
≥ 0, F (x, y) is increasing in x,
(iv) and yet, F(x,y) is not a valid joint CDF function. That is,
there is no random vector (X, Y ) whose joint CDF is F .

For continuous random variables X and Y with joint probability
density function. f(x,y) = xe−(x+y) when x > 0 and y
> 0 f(x,y) = 0 otherwise
a. Find the conditional density F xly (xly)
b. Find the marginal probability density function fX (x)
c. Find the marginal probability density function fY (y).
d. Explain if X and Y are independent

Consider the function f(x,y)=y+sin(x/y)
a) Find the equation of the tangent plane to the graph offat the
point(1,3)
b) Find the linearization of the function f at the point(1;3)and
use it to approximate f(0:9;3:1).
c) Explain why f is differentiable at the point(1;3)
d)Find the differential of f
e) If (x,y) changes from (1,3) to (0.9,3.1), compare the values
of ‘change in f’ and df

Suppose that the joint probability density function of the
random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤
x ≤ 1, 0 ≤ y ≤ 1 0 otherwise.
(a) Sketch the region of non-zero probability density and show
that c = 3/ 2 .
(b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1).
(c) Compute the marginal density function of X and Y...

(1) (5 pts) Consider the function f : + ×+ → given by f(x, y) =
x! y!(x−y)! . Where and x and y are positive integers h Hint: this
is the combination formula, x y i (a) What types of relationships
are generated by this function, please justify your answers using
examples or counter examples. (b) How many combinations of 2 pairs
can be generated from a power of R, assuming there are 4 element in
set R .

X and Y are continuous random variables. Their joint probability
density function is given as f(x,y) = 1/5 (y+2) for 0<y<1 and
y-1<x<y+1. Calculate the conditional expectation
E(x/y=0).
Please show all the work and explain if the answer will be a
number or just y in a given range.

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