Download PROBABILITY AND QUEUEING THEORY ASSIGNMENT

PROBABILITY AND QUEUEING THEORY
ASSIGNMENT - I
1. A r.v X has the following probability function.
X:0
1
2
3
4
5
6
P(X): 0
k
2k
2k
3k
k2
2k2
7
7k2+k
Find k (ii) Evaluate P(X<6),P(X
) and P(0<X<5) (iii) If P(X
the minimum value of k and determine the distribution function of X.
)> , find
2. Given the following probability distribution of X compute
(i)E(X)
x : -3
P(x) : 0.05
3.
(ii)E(X2)
(iii)E(2X
) (iv) Var(2X
)
-2
-1
0
1
2
0.10
0.30
0
0.30
0.15
If a r.v X has the pdf
3
0.10
f(x)=
Find the constant ‘c’ and mean of X.
4. If a r.v X has the pdf
Find (i) P(X>3)
f(x)=
(ii) MGF of X
(iii) E(X) and Var(X).
5. The probability function of an infinite discrete distribution is given by
P(X=x) =
, x = 1, 2,3,……….Find the mean and variance of the
distribution also find P(X is even),P(X
) and P(X is divisible by 3).
PROBABILITY AND QUEUEING THEORY
ASSIGNMENT – II
1. In a large consignment of electric bulbs 10% are defective. A random sample of 20 is
taken for inspection. Find the probability that
(i)
All are goods bulbs
(ii)
At-most there are 3 defectives bulbs
(iii)
Exactly there are 3 defective bulbs.
2. Let X be a Poisson r.v with parameter
.Show that
(i) P(X is even) =
(ii) P(X is odd) =
3. Let X1 , X2 be independent r.v’s each having geometric distribution qpk , k = 0,
1,2,…… Show that the conditional distribution of X1 given X1 + X2 is uniform.
4. A fast food chain finds that the average time customers have to wait for service is 45
seconds. IF the waiting time can be treated as an exponential random variable, what is the
probability that a customer will have to wait more than 5 minutes given that already he
waited for 2 minute?.
5. Suppose that the lifetime of a certain kind of an emergency backup battery(in hrs) is a
r.v X having the Weibull distribution
Find the mean lifetime of these batteries.
The probability that such a battery will last more than 300 hrs.
PROBABILITY AND QUEUEING THEORY
ASSIGNMENT – III
1. Let X & Y have the jpdf f(x.y)=2,0<x<y<1. Find the marginal density function of Y
given X=x.
2. If the jpdf of a two dimensional r.v. (X,Y) is given by f(x,y) is given by , f(x,y) =
x2+(xy/3), 0<x<1; 0<y<2. Find i) P(X>1/2) ; ii) P(X<Y)
and iii) P(Y<1/2 / X<1/2). Check whether the conditional density functions are valid.
3. If the jpdf of a two dimensional r.v. (X,Y) is given by f(x,y) is given by , f(x,y) = 8xy ,
0<x<1; 0<y<x. Find P(Y<1/2 / X< 1/2).
Also find the conditional density function f(y/x).
4. The two dimensional r.v. (X,Y) has the jpdf f(x,y) = (x+2y)/27 , x = 0,1,2 ; y= 0,1,2.
Find the conditional distribution of Y given X=x. Also find the conditional distribution of
X given Y = 1.
5. The joint probability mass function of X and Y is
0
1
2
0
0.1
0.04
0.02
1
0.08
0.20
0.06
2
0.06
0.14
0.30
Y
X
PROBABILITY AND QUEUEING THEORY
ASSIGNMENT – IV
1. Two rvs have the Joint pdf f ( x , y) =
Find
i)
The Correlation coefficient
ii)
The two regression lines
iii)
The two regression curves for means.
2. Let X and Y have the following joint probability mass function.
X
-1
1
Y
0
1/8 3/8
1
2/8 2/8
Find the Correlation coefficient of X and Y.
3. The regression equations of X and Y on X are respectively 4 x-5y + 22 = 0 and
20x-9y = 107 and variance of x = 25.Find the means of X and Y and also find
correlation coefficient.
4. Find the correlation coefficient for the data:
X
80
45
55
56
58
60
65
68
70
75
85
Y
82
56
50
48
60
62
64
65
70
74
90
5. Given the Joint pdf of X and Y as f( x , y) =
distribution of U = X + Y.
. Find the
PROBABILITY AND QUEUEING THEORY
ASSIGNMENT - V
1. Show that the process X (t )  A cos t  B sin t where A and B are random variables )
is Wide- Sense Stationary, if i) E(A)=E(B)=0
E(AB)=0.
ii) E(A2)=E(B2)
and iii)
2. If X (t )  sin( t  Y ) , where Y is uniformly distributed in (0,2  ), prove that {X(t)}is
a WSS.
3. A fair die is tossed repeatedly. If Xn denotes the maximum of the numbers occurring in
the first n tosses, find the tpm P of the Markov Chain{Xn}.
2 1

0

3 3

1
1
4. The three state Markov chain is given by the tpm P  
. Prove that the
0
2
2
1 1

0

2 2

chain is irreducible and all the states are aperiodic and non – null persistent . Find also
the steady – state distribution of the chain.
5.Patients arrive randomly and independently at a doctor’s consulting room from 8 A.M
at an average rate of one in 5 min. The waiting room can hold 2 persons. What is the
probability that the room will be full when doctor arrives at 9 A.M?.
Motivation Study- I
Find P0 and Pn values for the Models (M/M/1):(  /FIFO), (M/M/s):(  /FIFO),
(M/M/1):( k/FIFO) and (M/M/s):( k/FIFO).
Motivation Study- II
1.Find P0 and Pn values for the Model (M/G/1):(  /GD).
2. Write short note about Open and Closed Network.