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NYU Polytechnic School of Engineering
MA 3022 SPRING 2014
WORKSHEET II
Print Name:
Signature:
Section:
Instructor: Dr. Manocha
Date:
ID #:
Directions: Complete all questions clearly and neatly. You must show all work to have
credit. Unclear work will not be graded.
Problem Possible
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Total
100
Points
Your signature:
(1)(10 points)
Suppose that A, B, C, are independent random variables, each being uniformly distributed
over (0, 1).
(a) What is the joint cumulative distribution function of A, B, C?
(b) What is the probability that all of the roots of the equation Ax2 + Bx + C = 0 are real?
Your signature:
(2)(10 points)
The joint density function of X and Y is given by
f (x, y) = xe−x(y+1)
x > 0, y > 0
Find the conditional density of X, given Y = y, and that of Y , given X = x.
Your signature:
(3)(10 points)
If X and Y are independent and identically distributed uniform random variables on (0,1),
compute the joint density of
(a) U = X + Y, V = X/Y ;
(b) U = X, V = X/Y ;
(c) U = X + Y, V = X/(X + Y )
Your signature:
(4) (10 points)
Repeat the previous problem when X and Y are independent exponential random variable,
each with parameter λ = 1.
Your signature:
(5) (10 points)
If X, Y and Z are independent random variables having identical density functions f (x) = e−x ,
0 < x < ∞, derive the joint distribution of U = Y + Z, V = X + Z, W = X + Y .
Your signature:
(6) (10 points)
If X and Y have joint density function
f(x,y)=
find
(a) E[XY ].
(b) E[X]
(c) E[Y ].
1/y,
0,
if 0 < y < 1, 0 < x < y
otherwise
Your signature:
(7)(10 points)
If E[X] = 1 and Var(X) = 5, find
(a) E[(2 + X)2 ].
(b) Var(4 + 3X).
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(8)(10 points)
The random variables X and Y have a joint density function given by
f(x,y)=
Compute Cov(X, Y ).
2e−2x /x
0
0 ≤ x < ∞, 0 ≤ y ≤ x
otherwise
Your signature:
(9)(10 points)
Suppose that Xi , i = 1, 2, 3 are independent Poison random variables with respective means
λi , i = 1, 2, 3. Let X = X1 + X2 and Y = X2 + X3 . The random vector X, Y is said to have
a bivariate Poisson distribution.
(a) Find E[X] and E[Y ].
(b) Find Cov(X, Y ).
(c) Find the joint probability mass function P {X = i, Y = j}.
Your signature:
(10)(10 points)
The joint density function of X and Y is given by
1
f (x, y) = e(−y+x/y) ,
y
Find E[X], E[Y ], and show that Cov(X, Y ) = 1.
x > 0, y > 0