EC 221 Probability and Random Processes Assignment-I
1. Given S cfw_1, 2,3 findtwofieldscontainingthesubsetcfw_2.
2. Suppose isafieldofsubsetsofasetSand B .Showthat C cfw_ A B | A isa
fieldofthesubsetsofB.
3. Suppose S (thesetofnaturalnumbers)and C cfw_
Tutorial 5
1. Suppose X is a binomial random variable with the PMF
()
p X ( k ) = n p k (1 p ) n k
k
0 p 1, k = 0,1,., n
(a) Find the most likely value(s) of X.
(b) Find a condition on n and p so that
(c) p X (k ) = p X (n k ) k = 0,1,., n
(d) If p=0.3, f
EC 221:Probability and Random Processes
Assignment-3
1. A well shuffled deck of 52 cards is distributed among 4 players North, East, South and
West. What are the probabilities that
(a) North has 2 kings,
(b) West has 4 kings, 4 aces and 4 queens.
(c) Sout
! " #$%
"#$
#
D( f | g ) =
&!
f ( x)log ( f ( x) / g ( x)dx
'!
+
,- !
( '!
(.
I A ( s) =
( /
&! * (
.
1 ( (
.
.
1 if s A
0 otherwise
&
2
,
2+
*
( /
&
X
/
( a>03
P(cfw_ X + a)
4
&
I A B = I A I B = min( I A , I B )
!+
0
D ( f | g ) 0
) * +
2
+ a2
2
Consi
Tutorial 8
1.
9/3/11
(a) ForrandomvariablesXandY,thejointPDFisgivenby
1
, x 1, y 1
f X ,Y ( x, y ) 4
0 otherwise
Show that XandYareindependentrandomvariables.
(b) Suppose XandYaredistributedas
1
2
2
, x y 1
f X ,Y ( x, y )
0 otherwise
ExamineifXandYare
EC 221:Probability and Random Processes
Assignment-2
1. Consider the sequence cfw_ An given by
An ( a
1
n 1
, b n1 ), a, b
Show that lim supcfw_ An lim infcfw_ An ( a, b] and hence lim An exists.
n
n
n
2.Define the Borel sigma algebra as the minim
Tutorial 4
1. A player has equal probability of win in each game against his opponent. Which is more
porobable?
(a) Three wins from 5 games or 4 wins from 7 games?
(b) At least three wins from from 5 games or at least 4 wins from 7 games?
2. A player toss
Tutorial 11 30/3/11
1. The Poisson process
cfw_ X (t ), t 0 is
a random
process where
the increments
X (t j ) X (ti ), t j > ti are independent Poisson random variables with the rate parameter
(t2 t1 ) .
(a) Find p X ( t1 ), X ( t2 ) (k1 , k2 ) for t1 >
Tutorial9Date16/3/11
1. Suppose X and Y denote the independent random variables with the identical density
function f. Find the expressions for the PDF of (i) Z = min( X , Y ) and (ii)
Z = max( X , Y )
2. Suppose X ~ exp(1) and Y ~ exp(1) are two RVs and
Tutorial 6
1. (a) Suppose X is a discrete random variable with RX = cfw_0,1,. show that
EX = P (cfw_ X n)
n =0
(b) Suppose X is a non-negative continuous random variable. Show that
EX = (1 FX ( x) dx
0
2. (a) Let X be a continuous random variable with
1
f