Question

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).

Answer #1

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 the probability density function of the random variable X be
f(x) = { e ^2x if x ≤ 0 ;1 /x ^2 if x ≥ 2 ; 0 otherwise}
Find the cumulative distribution function (cdf) of X.

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).

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).

2. Let the probability density function (pdf) of random variable
X be given by:
f(x) = C (2x -
x²),
for
0< x < 2,
f(x) = 0,
otherwise
Find the value of
C.
(5points)
Find cumulative probability function
F(x)
(5points)
Find P (0 < X < 1), P (1< X < 2), P (2 < X
<3)
(3points)
Find the mean, : , and variance,
F².
(6points)

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 X be a continuous random variable with the probability
density function f(x) = C x, 6 ≤ x ≤ 25, zero otherwise.
a. Find the value of C that would make f(x) a valid probability
density function. Enter a fraction (e.g. 2/5): C =
b. Find the probability P(X > 16). Give your answer to 4
decimal places.
c. Find the mean of the probability distribution of X. Give your
answer to 4 decimal places.
d. Find the median...

Let the probability function of the random variable X be f(x) =
{ x⁄45 if x = 1, 2, 3, ⋯ ⋯ ,9 ; 0 otherwise}
Find E(X) and Var(X)

The joint probability density function of two random variables
(X and Y) is given by fX,Y (x, y) = ( C √y (y ^(α+1)) exp {( −
y(2β+x ^2 ) )/2 } , x ∈ (−∞,∞), y ∈ [0,∞), 0 otherwise. (a) Find C.
(b) Find the marginal density of Y . What type of distribution does
Y follow? (c) Find the conditional density of X | Y . What type of
distribution is this?

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)=

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