Question

Is
the function continuous on (-infinity, infinity)? Explain.

f(x) = cos x; if x<0

0; if x= 0

1-x^2 ; if x>0

Answer #1

If f(x) is a probability density function of a continuous random
variable, then f(x)=?
a-0
b-undefined
c-infinity
d-1

Show that the function f(x)=x2sin(x) is uniformly
continuous on [0,b] for any constant b>0, but that is not
uniformly continuous on [0,infinity)

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.

if f''(x)= -cos(x)+sin(x), and f(0)=1 and f(pi)=), what is the
original function

X is a continuous random variable with the cumulative
distribution function
F(x) = 0
when x <
0
= x2
when 0 ≤ x ≤
1
= 1
when x >
1
Compute P(1/4 < X ≤ 1/2)
What is f(x), the probability density function of X?
Compute E[X]

For the given function: f [0; 3]→R is continuous, and all of its
values are rational numbers. It is also known that f(0) = 1. Can
you find f(3)?Justification b) Let [x] denote the smallest integer,
not larger than x. For instance,[2.65] = 2 = [2], [−1.5] =−2.
Caution: [x] is not equal to |x|! Find the points at which the
function f : R→R. f(x) = cos([−x] + [x]) has or respectively, does
not have a derivative.

6. A continuous random variable X has probability density
function
f(x) =
0 if x< 0
x/4 if 0 < or = x< 2
1/2 if 2 < or = x< 3
0 if x> or = 3
(a) Find P(X<1)
(b) Find P(X<2.5)
(c) Find the cumulative distribution function F(x) = P(X< or
= x). Be sure to define the function for all real numbers x. (Hint:
The cdf will involve four pieces, depending on an interval/range
for x....

consider the function
f(x)= -1/x, 3, √x+2
if x<0
if 0≤x<1
if x≥1
a)Evaluate lim,→./ f(x) and lim,→.2 f(x)
b. Does lim,→. f(x) exist? Explain.
c. Is f(x) continuous at x = 1? Explain.

Find the value of C > 0 such that the function
?C sin2x, if0≤x≤π,
f(x) =
0, otherwise
is a probability density function.
Hint: Remember that sin2 x = 12 (1 − cos 2x).
2. Suppose that a continuous random variable X has probability
density function given by the above f(x), where C > 0 is the
value you computed in the previous exercise. Compute E(X).
Hint: Use integration by parts!
3. Compute E(cos(X)).
Hint: Use integration by substitution!

A function f is said to be continuous on the _______ at x = c if
lim x → c + f ( x ) = f ( c ).
A function f is said to be continuous on the _______ at x = c if
lim x → c − f ( x ) = f ( c ).
A real number x is a _______ number for a function
f if f is discontinuous at x or f...

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