Department of Mathematics
MTL 106/MAL 250 (Probability and Stochastic Processes)
Tutorial Sheet No. 2
1. Do the following functions dene distribution functions.
cfw_
0, x < 0
1 ex , 0 < x <
1
1
1
x, 0 x 2
(a) F (x) =
(b) F (x) = ( )tan x, < x < (c) F (x

Department of Mathematics
MTL 106 (Introduction to Probability and Stochastic Processes)
Tutorial Sheet No. 8
Answer for Selected Problems
1. W - waiting time; = , P (W = 0) = 1 ,
P (W > t) = e(1)t , t 0; E(W ) = .
2. M/M/1 model with = 8 hour1 ,
(a) 1 0

Department of Mathematics
MTL 106 (Introduction to Probability and Stochastic Processes)
Tutorial Sheet No. 8
1. Consider a M/M/1 queuing model. Find the waiting time distribution for any customer in this model.
Deduce the mean waiting time from the above

Department of Mathematics
MTL 106 (Introduction to Probability and Stochastic Processes)
Tutorial Sheet No. 7
Answer for Selected Problems
Note: =
everywhere
1. P (t) = P (t)Q K.F.E with P (t) = (P0 (t), P1 (t), P2 (t) and
P (t) = QP (t) K.B.E with P (t)

Department of Mathematics
Tutorial Sheet No. 6
MTL 106 (Introduction to Probability and Stochastic Processes)
1. Let X(t) = A0 + A1 t + A2 t2 , where A s are uncorrelated random variables with mean 0 and variance 1. Find
i
the mean function and covariance

Department of Mathematics
Tutorial Sheet No. 5
MTL 106 (Introduction to Probability and Stochastic Processes)
1. Find E(Y /x) where (X, Y ) is jointly distributed with density
cfw_
y
y
1+x
, x, y 0
(1+x)4 e
f (x, y) =
0,
otherwise.
2. Let X have a beta d

Department of Mathematics
Tutorial Sheet No. 4
MTL 106 (Introduction to Probability and Stochastic Processes)
1. Let X and Y be independent random variables. The range of X is cfw_1, 3, 4 and the range of Y is cfw_1, 2.
Partial information on the probabil

Department of Mathematics
MTL 106 (Introduction to Probability and Stochastic Processes)
Tutorial Sheet No. 3
1. X has a uniform distribution over the set of integers cfw_n, (n 1), . . . , 1, 0, 1, . . . , (n 1), n. Find the
distribution of (i) |X| (ii) X

Department of Mathematics
MTL 106 (Introduction to Probability and Stochastic Processes)
Tutorial Sheet No. 2
Answer for selected Problems
1. a)No
b)No
c)Yes
cfw_ n1 n2
( n )( n ) . . . ( n(k2) )( k1 ), k = 2, 3, . . . , n + 1
n
n
2. P [X = k] =
0,
otherw

Department of Mathematics
MTL 106 (Introduction to Probability and Stochastic Processes)
Tutorial Sheet No. 1
1. Items coming o a production line are marked defective (D) or non-defective (N ). Items are observed and
their condition noted. This is continu

Department of Mathematics
MTL 106 (Introduction to Probability and Stochastic Processes)
Tutorial Sheet No. 1
Answer for selected Problems
1. | |= 12
= cfw_DD, N DD, N DN D, N N DD, N N DN, N N N N, N N N D, N DN N,
DN N N, DN DN, DN N D, DN DD
5. a) T
d