Question

Let X be a random variable with probability density function fX (x) = I (0, 1) (x). Determine the probability density function of Y = 3X + 1 and the density function of probability of Z = - log (X).

Answer #1

Let X be a random variable with the probability density function
fx(x) given by:
fx(x)=
1/4(2-x), 0<x<2
1/4(x-2), 2<=x<4
0, otherwise.
Let Y=|X-3|. Compute the probability density function of Y.

Let X be a continuous random variable with a probability density function
fX (x) = 2xI (0,1) (x) and let it be the function´
Y (x) = e^−x
a. Find the expression for the probability density function fY (y).
b. Find the domain of the probability density function fY (y).

Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { cx2y, 0 < x2 < y < x for
x > 0 0, otherwise }. compute the marginal probability density
functions fX(x) and fY (y). Are the random variables X and Y
independent?.

Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { e −x−y , 0 < x, y < ∞ 0,
otherwise } . a. Let W = max(X, Y ) Compute the probability density
function of W. b. Let U = min(X, Y ) Compute the probability
density function of U. c. Compute the probability density function
of X + Y .

Let X be a random variable with probability density function
fX(x) given by fX(x) = c(4 − x ^2 ) for |x| ≤ 2 and zero
otherwise.
Evaluate the constant c, and compute the cumulative distribution
function.
Let X be the random variable. Compute the following
probabilities.
a. Prob(X < 1)
b. Prob(X > 1/2)
c. Prob(X < 1|X > 1/2).

Let X be a random variable with probability density function
fX(x) = {c(1−x^2)if −1< x <1, 0 otherwise}.
a) What is the value of c?
b) What is the cumulative distribution function of X?
c) Compute E(X) and Var(X).

1. Let (X; Y ) be a continuous random vector with joint
probability density function
fX;Y (x, y) =
k(x + y^2) if 0 < x < 1 and 0 < y < 1
0 otherwise.
Find the following:
I: The expectation of XY , E(XY ).
J: The covariance of X and Y , Cov(X; Y ).

Let fX,Y be the joint density function of the random variables X
and Y which is equal to fX,Y (x, y) = { x + y if 0 < x, y <
1, 0 otherwise. } Compute the probability density function of X + Y
. Referring to the problem above, compute the marginal probability
density functions fX(x) and fY (y). Are the random variables X and
Y independent?

Let X have density fX(x) = C/ sqrt(x), 0 < x <
1
(a) Find the cumulative distribution FY (y) and
probability density function fY (y) of Y = X(1 − X).
(b) Find probability density function fZ(z) of Z =
X1/4 .
SHOW ALL STEPS OF WORKING OUT CLEARLY
PLEASE.THANKS!

Let X be a gamma random variable with parameters α > 0 and β
> 0. Find the probability density function of the random
variable Y = 3X − 1 with its support.

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