In: Statistics and Probability

Let (X, Y) be a random vector with a function of the joint density given by ˜ fX, Y (x, y) = k (2x + y) I (2,6) (x) I (0.5) (y)

a) Determine k so that f X, Y (x, y) is a true probability density function joint quality.

b) Determine the marginal probability density functions of X and Y.

c) Calculate P (3 <X <4, Y> 2).

d) Calculate P (X + Y> 4).

please note that I(0.5) is improper representation for defining range, so assuming the correction to be as I(0,5) the question has been solved.

Thanks!

Let (X, Y) be a random vector with a function of the joint density given by ˜
fX, Y (x, y) = k (2x + y) I (2,6) (x) I (0.5) (y)
a) Determine k so that f X, Y (x, y) is a true probability
density function joint quality.
b) Determine the marginal probability density functions of X and
Y.
c) Calculate P (3 <X <4, Y> 2).
d) Calculate P (X + Y> 4).

. Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { 1, 0 < x, y < 1 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
..

If the joint probability density function of the random
variables X and Y is given by f(x, y) = (1/4)(x + 2y) for 0 < x
< 2, 0 < y < 1, 0 elsewhere
(a) Find the conditional density of Y given X = x, and use it to
evaluate P (X + Y/2 ≥ 1 | X = 1/2)
(b) Find the conditional mean and the conditional variance of Y
given X = 1/2
(c) Find the variance...

Consider a continuous random vector (Y, X) with joint
probability density function
f(x, y) = 1
for 0 < x < 1, x < y < x + 1.
What is the marginal density of X and Y? Use this to compute
Var(X) and Var(Y)
Compute the expectation E[XY]
Use the previous results to compute the correlation Corr (Y,
X)
Compute the third moment of Y, i.e., E[Y3]

Consider a continuous random vector (Y, X) with joint
probability density function f(x, y) = 1 for 0 < x < 1, x
< y < x + 1.
A. What is the marginal density of X and Y ? Use this to compute
Var(X) and Var(Y).
B. Compute the expectation E[XY]
C. Use the previous results to compute the correlation Corr(Y,
X).
D. Compute the third moment of Y , i.e., E[Y3].

Let X and Y be two continuous random variables
with the joint probability density function of for 0 < x
< 2, 0 < y < 2, x + y < 1,where c is a
constant. (In all the following answers, you do NOT need to find
what the value of c is; just treat it as a number.)
(a) Write out the marginal distribution of Y.
(b) P(Y < 1/3) = ?
(c) P(X < 1.5, Y < 0.5)=...

The joint probability density function for two random variables
X and Y is given as, fx,y (x, y) = (2/3)(1 + 2xy3 ), 0
< x < 1, 0 < y < 1
(a) Find the marginal probability density functions for X and Y
.
(b) Are X and Y independent? Justify your answer.
(c) Show that E[X] = 4/9 and E[Y ] = 7/15 .
(d) Calculate Cov(X, Y )

The joint probability density function (PDF) of two random
variables (X,Y) is given by
???(?,?) = { 1, 0 ≤ ? ≤ 2,0 ≤ ? ≤ 1,2? ≤ ? 0, otherwise
1) Find the correlation coefficient ??? between the two random
variables X and Y
Find the probability P(Y>X/2).
help please asap

Suppose that the joint probability density function of ˜ (X, Y) is given by:´
f X,Y (x,y) = 4x/y3 I(0.1)(x), I (1, ∞)(y).
Calculate
a) P(1/2 < X < 3/4, 0 < Y ≤ 1/3).
b) P(Y > 5).
c) P(Y > X).

The joint density function for random variables X,
Y, and Z is
f(x, y,
z)= Cxyz if 0 ≤
x ≤ 1, 0 ≤ y ≤ 2, 0 ≤
z ≤ 2, and
f(x, y,
z) = 0 otherwise.
(a) Find the value of the constant C.
(b) Find P(X ≤ 1, Y ≤ 1, Z ≤ 1).
(c) Find P(X + Y + Z ≤ 1).

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