Question

Assume that X and Y has a continuous joint p.d.f. as
(**28x^2)*(y^3)** in **0<y<x<1**
interval. Otherwise the joint p.d.f. is equal to 0.

- Prove that the mentioned f(x,y) is a joint probability density function.
- Calculate
**E(X)** - Calculate
**E(Y)** - Calculate
**E(****X****2****)** - Calculate
**Var(X)** - Calculate
**E(XY)** - Calculate
**P(X< 0.1)** - Calculate
**P(X> 0.1)** - Calculate
**P(X>2)** - Calculate
**P(-2<X<0.1)**

Answer #1

I need solution and formula as clear.Otherwise , I will
have to report the answer.
Assume that X and Y has a continuous joint p.d.f. as
28x2y3
in 0<y<x<1 interval. Otherwise the joint
p.d.f. is equal to 0.
Prove that the mentioned f(x,y) is a joint probability density
function.
Calculate E(X)
Calculate E(Y)
Calculate
E(X2)
Calculate E(XY)

Let X and Y are two continuous random variables. It's joint
p.d.f is given as:
f(x,y) = 2 , 0 < x < y < 1
= 0, otherwise
Calculate P(x+y >1)

Given the following joint density,
f(x,y)={10xy^2 if 0<x<y<1
f(x,y)={ 0 otherwise
1. frequency function x given y
2. E(x given y), Var(x given y)
3. Var(E(x given y), E(Var(x given y)

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 ).

Consider the joint pdf
f(x, y) = 3(x^2+ y)/11
for 0 ≤ x ≤ 2, 0 ≤ y ≤ 1.
(a) Calculate E(X), E(Y ),
E(X^2), E(Y^2), E(XY
), Var(X), Var(Y ),
Cov(X, Y ).
(b) Find the best linear predictor of Y given
X.
(c) Plot the CEF and BLP as a function of X.

X and Y are continuous random variables. Their joint probability
distribution function is :
f(x,y) = 1/5(y+2) , 0 < y < 1, y-1 < x < y +1
= 0, otherwise
a) Find marginal density of Y, fy(y)
b) Calculate E[X | Y = 0]

The random variables X and Y have a joint p.d.f. given by
f(x,y) = (3(x +y −xy))/7 for 0 ≤ x ≤ 1 and
0 ≤ y ≤ 2. Find the following.
(a) E[X], E[Y ]
(b) Cov[X,Y]

Suppose X and Y are continuous random variables with joint pdf
f(x,y) = 2(x+y) if 0 < x < < y < 1 and 0 otherwise.
Find the marginal pdf of T if S=X and T = XY. Use the joint pdf of
S = X and T = XY.

A continuous random variable X has the following
probability density function F(x) = cx^3, 0<x<2 and 0
otherwise
(a) Find the value c such that f(x) is indeed
a density function.
(b) Write out the cumulative distribution function of
X.
(c) P(1 < X < 3) =?
(d) Write out the mean and variance of X.
(e) Let Y be another continuous random variable such
that when 0 < X < 2, and 0 otherwise. Calculate
the mean of Y.

The joint probability density function of x and y is given by
f(x,y)=(x+y)/8 0<x<2, 0<y<2 0 otherwise
calculate the variance of (x+y)/2

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