ECEN/MAE 5513
Stochastic Systems
Fall 2015
Homework Assignment #7
Problem 1: Statistical independent random variables X and Y have probability densities
3 (4 x 2 ), 2 x 2
f X ( x) 32
elsewhere
0,
f Y ( y ) 12 u ( y 1) u ( y 1)
Find the exact probability
ECEN 5513
Stochastic Systems
Fall 2015
Homework Assignment #2
Problem 1: At a military installation, six similar radars are placed in operation. It is known that a
radars probability of failing to operate before 500 hours of on time have accumulated is 0.
ECEN/MAE 5513
Stochastic Systems
Fall 2015
Homework Assignment #4
Problem 1: For the binomial density function
N
N
f X ( x) p k (1 p) N k ( x k ) ,
k 0 k
show that E ( X ) Np and X2 Np(1 p ) .
Problem 2: For the Poisson density function
bk
f X ( x ) e b
ECEN 5513
Stochastic Systems
Fall 2015
Homework Assignment #3
Problem 1: For real constants b 0 , c 0 , and any a, find a condition on constant a and a
relationship between c and a (for given b) such that the function
a[1 ( x / b)], 0 x c
f X ( x)
elsewh
ECEN/MAE 5513
Stochastic Systems
Fall 2015
Homework Assignment #6
Problem 1: A joint sample space for two random variables X and Y has four elements (1,1),
(2,2), (3,3) and (4,4). Probabilities of these elements are 0.1, 0.35, 0.05 and 0.5 respectively.
a
ECEN 5513
Stochastic Systems
Fall 2015
Homework Assignment #1
Problem 1: Sketch a Venn diagram for three events where A B , B C , C A ,
but A B C .
Problem 2: Use De Morgans laws to show that
a) A ( B C ) ( A B ) ( A C ) ;
b) ( A B C ) A B C .
Problem 3:
ECEN/MAE 5513
Stochastic Systems
Fall 2015
Homework Assignment #8
Problem 1: Discrete random variables X and Y have the joint density
f XY ( x, y ) 0.4 ( x ) ( y 2) 0.3 ( x ) ( y 2) 0.1 ( x ) ( y ) 0.2 ( x 1) ( y 1)
Determine the value of , if any, that m
ECEN/MAE 5513
Stochastic Systems
Fall 2015
Homework Assignment #9
Problem 1: A radom process X (t ) has periodic sample functions as shown below, where B, T
and 4t 0 T are constants but is a random variable uniformly distributed on the interval
(0, T ) .
ECEN/MAE 5513
Stochastic Systems
Fall 2015
Homework Assignment #5
Problem 1: The Laplace density function
1 xm / b
f X ( x)
e
2b
has a characteristic function
e jm
.
X ( )
1 (b ) 2
Use this characteristic function to find the mean and variance of the r
ECEN/MAE 5513
Stochastic Systems
Fall 2015
Homework Assignment #10
Problem 1: Let A and B be two random variables. We form the random process
X (t ) A cos( 0 t ) B sin( 0 t )
where 0 is areal constant.
a) Show that if A and B are uncorrelated with zero me