Question

Suppose that the random variables X and Y
have the following joint probability density function.f (x, y) =
ce^{−5x − 7y}, 0
< y < x. |

(a) | Find the value of c. |

(b) | Find P(X < 1/3 , Y <
2) |

Answer #1

a)

for this to be valid:

f(x,y) dy dx must be 1

f(x,y)
dy dx =c*(e^{-5x}^{-7y})
dy dx =c*e^{-5x}*(-e^{-7y}/7)|^{x}_{0}
dx

=
(c/7)e^{-5x}(1-e^{-7x}) dx
=(c/7)*(-e^{-5x}/5+e^{-12x}/12) |^{}_{0}
=(c/7)*(1/5-1/12)=(c/7)*(7/60)=1

**c =60**

b)

P(X<1/3,Y<2)=f(x,y)
dy dx = (60/7)*(-e^{-5x}/5+e^{-12x}/12)
|^{1/3}_{0} =**0.689296**

Suppose that the random variables X and Y
have the following joint probability density function.
f (x, y) =
ce−9x − 7y, 0
< y < x.
(a)
Find the value of c.
(b)
Find P(X <1/6
, Y < 1)

Suppose that the joint probability density function of the
random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤
x ≤ 1, 0 ≤ y ≤ 1 0 otherwise.
(a) Sketch the region of non-zero probability density and show
that c = 3/ 2 .
(b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1).
(c) Compute the marginal density function of X and Y...

Suppose X, Y, and Z are random
variables with the joint density function
f(x, y, z) = Ce−(0.5x + 0.2y + 0.1z) if x ≥ 0, y ≥ 0,
z ≥ 0, and f(x, y, z) = 0 otherwise.
(a) Find the value of the constant C.
(b) Find P(X ≤ 0.75 , Y ≤ 0.5).
(Round answer to five decimal places).
(c) Find P(X ≤ 0.75 , Y ≤ 0.5 ,
Z ≤ 1). (Round answer to six decimal...

a) The joint probability density function of the random
variables X, Y is given as
f(x,y) =
8xy
if 0≤y≤x≤1 , and 0
elsewhere.
Find the marginal probability density functions.
b) Find the expected values EX and
EY for the density function above
c) find Cov X,Y .

7. Suppose that random variables X and Y have a joint density
function given by: f(x, y) = ? + ? 0 ≤ ?≤ 1, 0 ≤ ? ≤ 1
(a) Find the density functions of X and Y, f(x) and f(y).
(b) Find E[X] and Var(Y).

A joint density function of the continuous random variables
x and y is a function f(x,
y) satisfying the following properties.
f(x, y) ≥ 0 for all (x, y)
∞
−∞
∞
f(x, y) dA = 1
−∞
P[(x, y) R] =
R
f(x, y) dA
Show that the function is a joint density function and find the
required probability.
f(x, y) =
1
8
,
0 ≤ x ≤ 1, 1 ≤ y ≤ 9
0,
elsewhere
P(0 ≤...

Let X and Y be two continuous random variables with joint
probability density function
f(x,y) =
6x 0<y<1, 0<x<y,
0 otherwise.
a) Find the marginal density of Y .
b) Are X and Y independent?
c) Find the conditional density of X given Y = 1 /2

For continuous random variables X and Y with joint probability
density function. f(x,y) = xe−(x+y) when x > 0 and y
> 0 f(x,y) = 0 otherwise
a. Find the conditional density F xly (xly)
b. Find the marginal probability density function fX (x)
c. Find the marginal probability density function fY (y).
d. Explain if X and Y are independent

Suppose X and Y are continuous random variables with joint
density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
(a). Compute the joint CDF F(x,y).
(b). Compute the marginal density for X and Y .
(c). Compute Cov(X,Y ). Are X and Y independent?

Let X and Y be two continuous random variables with joint
probability density function f(x,y) = xe^−x(y+1), 0 , 0< x <
∞,0 < y < ∞ otherwise
(a) Are X and Y independent or not? Why?
(b) Find the conditional density function of Y given X = 1.(

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 32 minutes ago

asked 49 minutes ago

asked 51 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago