Question

Suppose that X is continuous random variable with PDF f(x) and CDF F(x). (a) Prove that if f(x) > 0 only on a single (possibly infinite) interval of the real numbers then F(x) is a strictly increasing function of x over that interval. [Hint: Try proof by contradiction]. (b) Under the conditions described in part (a), find and identify the distribution of Y = F(x).

Answer #1

Given pdf and CDF . Now the property of pdf is

.

Now

That is is a strictly increasing function of over the interval .

Given the transformation .

The pdf of is

The distribution of uniform in the interval

If X is a continuous random variable with pdf f(x) on the
interval [a,b] then show that a<E(X)<b.

Suppose X and Y are continuous random variables with joint
pdf
f(x,y) = x + y, 0 < x< 1, 0 < y< 1. Let W =
max(X,Y). Find EW.

Suppose the pdf of a random variable X is defined as:
f(x) = (x/16) + (1/4)
for -4 < x <= 0, and
f(x) = -((x^2)/36) + (1/4)
Find the cdf of X.

19. Let X and Y be continuous random variables with joint pdf:
f(x, y) = x−y for 0 ≤ y ≤ 1 and 1 ≤ x ≤ 2. If U = XY and V = X/Y ,
calculate the joint pdf of U and V , fUV (u, v).

a) Suppose that X is a uniform continuous random variable where
0 < x < 5. Find the pdf f(x) and use it to find P(2 < x
< 3.5).
b) Suppose that Y has an exponential distribution with mean 20.
Find the pdf f(y) and use it to compute P(18 < Y < 23).
c) Let X be a beta random variable a = 2 and b = 3. Find P(0.25
< X < 0.50)

Consider continuous random variables X and Y whose joint pdf is
f(x, y) = 1 with 0 < y < 1 − |x|. Show that Cov(X, Y ) = 0
even though X and Y are dependent. Note: For this problem, you only
need to show that the covariance is zero. You need not show that X
and Y are dependent.

Consider continuous random variables X and Y whose joint pdf is
f(x, y) = 1 with 0 < y <1 – abs(x). Show that Cov(X, Y ) = 0
even though X and Y are dependent. Note: For this
problem, you only need to show that the covariance is zero. You
need not show that X and Y are dependent.

Suppose that X is a continuous random variable with a
probability density function that is a positive constant on the
interval [8,20], and is 0 otherwise.
a. What is the positive constant mentioned
above?
b. Calculate P(10?X?15).
c. Find an expression for the CDF FX(x).
Calculate the following values.
FX(7)=
FX(11)=
FX(30)=

Identify the FALSE statement relative to a continuous random
variable, x, and its probability distribution:
Select one:
a. The distribution of x is modelled by a smooth curve
called a probability density function.
b. P(x = a) is given by the height of the density
function above the point, a.
c. The total area under the graph of the density function is
1.
d. The area above an interval and below the density curve gives
the probability of x lying...

Suppose the random variable (X, Y ) has a joint pdf for the
form
?cxy 0≤x≤1,0≤y≤1 f(x,y) = .
0 elsewhere
(a) (5 pts) Find c so that f is a valid distribution.
(b) (6 pts) Find the marginal distribution, g(x) for X and the
marginal distribution for Y , h(y).
(c) (6 pts) Find P (X > Y ).
(d) (6 pts) Find the pdf of X +Y.
(e) (6 pts) Find P (Y < 1/2|X > 1/2).
(f)...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 months ago

asked 2 minutes ago

asked 6 minutes ago

asked 7 minutes ago

asked 7 minutes ago

asked 7 minutes ago

asked 3 weeks ago

asked 10 minutes ago

asked 10 minutes ago

asked 11 minutes ago

asked 11 minutes ago

asked 11 minutes ago