MA 2261 Probability and Random process IV Sem ECE – S.SARULATHA Asst.Prof./MATHS 1 MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI – 621213 QUESTION BANK DEPARTMENT: ECE SEMESTER: IV SUBJECT CODE / Name: MA 2261/PROBABILITY AND RANDOM PROCESS UNIT – I: RANDOM VARIABLES PART -A (2 Marks) 1. The CDF of a continuous random variable is given by 5 0 , 0, () , 1 ,0 x x Fx X e x Find the pdf and mean of (AUC Nov/Dec 2011) (AUC Apr/May 2011) 2. The probability that a man shooting a target is 1/4 . How many times must he fire so that the probability of his hitting the target atleast once is more than 2/3? (AUC May/Jun 2012) 3. Find C, if 2 ; 1, 2, .... 3 n PX n c n (AUC May/Jun 2012) 4. A continuous random variable X has probability density function 2 3 ,0 1 () 0 , x x fx otherwise .Find K such that 0.5 PX k (AUC Nov/Dec 2010) 5. If X is uniformly distributed in ( , ) 2 2 . Find the probability distribution function of = (AUC Nov/Dec 2010) 6. Establish the memory less property of the exponential distribution. (AUC Apr/May 2011) 7. If the probability density function of a random variable X if () 0 2, ( 1.5 / 1). 2 x fx in x PX X find (AUC Apr/May 2010)
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MA 2261 Probability and Random process IV Sem ECE – S.SARULATHA Asst.Prof./MATHS
1
MAHALAKSHMI ENGINEERING COLLEGE
TIRUCHIRAPALLI – 621213
QUESTION BANK
DEPARTMENT: ECE SEMESTER: IV
SUBJECT CODE / Name: MA 2261/PROBABILITY AND RANDOM PROCESS
UNIT – I: RANDOM VARIABLES
PART -A (2 Marks)
1. The CDF of a continuous random variable is given by
5
0 , 0,( ) ,
1 ,0x
xF x X
e x
Find the pdf and meanof
(AUC Nov/Dec 2011)
(AUC Apr/May 2011)
2. The probability that a man shooting a target is 1/ 4 . How many times must he fire so
that the probability of his hitting the target atleast once is more than 2 / 3?
(AUC May/Jun 2012)
3. Find C, if 2
; 1,2,....3
n
P X n c n
(AUC May/Jun 2012)
4. A continuous random variable X has probability density function
23 ,0 1( )
0 ,
x xf x
otherwise
.Find K such that 0.5P X k
(AUC Nov/Dec 2010)
5. If X is uniformly distributed in ( , )2 2
. Find the probability distribution function of
𝑌 = 𝑡𝑎𝑛 𝑋 (AUC Nov/Dec 2010)
6. Establish the memory less property of the exponential distribution.
(AUC Apr/May 2011)
7. If the probability density function of a random variable X if
( ) 0 2, ( 1.5 / 1).2
xf x in x P X X find (AUC Apr/May 2010)
MA 2261 Probability and Random process IV Sem ECE – S.SARULATHA Asst.Prof./MATHS
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8. If the MGF of a uniform distribution for a random variable 1 5 4 . ( ).X is e e E xt
t tfind
(AUC Apr/May 2010)
9. If x is a normal random variable with mean zero and variance 2 , find the PDF of
Y e x (AUC Nov/Dec 2011)
10. The moment generating function of a random variable X is given by
3( 1)( ) . [ 0]?teM t e P X Whatis (AUC Nov/Dec 2012)
11. An experiment succeeds twice as often as it fails. Find the chance that in the next 4
trials, there shall be at least one success. (AUC Nov/Dec 2012)
PART –B (16 Marks)
1. The probability function of an infinite discrete distribution is given by
1
( 1,2,3....)2
P X x x x Find
(i) The value of X
(ii) P[X is even]
(iii) P[X is divisible by 3] (AUC Nov/Dec 2011)
2. A continuous random variable X has the PDF 2,
( ) 1
0 ,
kx
f x x
otherwise
Find a) the value of k
b) Distribution function of X
c) 0P X (AUC Nov/Dec 2011)
3. Let X and Y be independent normal variates with mean 45 and 44 and standard
deviation 2 and 1.5 respectively. What is the probability that randomly chosen values
of X and Y differ by 1.5 or more? (AUC Nov/Dec 2011)
4. If X is a uniform random variable in the interval (-2,2) find the probability density
function ( )Y X and E Y
(AUC Nov/Dec 2011)
5. A random variable X has the following probability distribution
2 2 2 2
0 1 2 3 4 5 6 7
( ) 0 2 2 3 2 7
x
p x k k k k k k k k
Find, (i) The value of k
(ii) (1.5 4.5 / 2)P X X and (iii) The smallest value of n for which
MA 2261 Probability and Random process IV Sem ECE – S.SARULATHA Asst.Prof./MATHS
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1( )
2P X n (AUC Nov/Dec 2010)
(AUC May/Jun 2012)
6. Find the MGF of the random variable X having the probability density function
3, 0
( ) 4
0 ,
e xf x
-x2
elsewhere
Also deduce the first four moments about the origin.
(AUC Nov/Dec 2010)
7. If X is the uniformly distributed in (-1,1),then find the probability density function of
sin2
xY
(AUC Nov/Dec 2010)
8. If X and Y are independent random variables following
(8,2) (12,4 3)
2 2 2
N N
P X Y P X Y
and respectively,findthevalueof suchthat(AUC Nov/Dec 2010)
9. The probability mass function of random variable X is defined as
2 2( 0) 3 , ( 1) 4 10 , ( 2) 5 1, 0,
( ) 0 0,1,2.
( )
( ) (0 2 / 0)
1( ) ( )
2
P X c P X c c P X c Wherec and
P X r if r
i
ii P X X
iii X F X
)Find
The valueof c.
Thelargest valueof for which
(AUCApr / May2010
10. If the probability that an applicant for a driver’s license will pass the road test on any
given trial is 0.8. What is the probability that he will finally pass the test
(i) On the fourth trial and
(ii) In less than 4 trials? (AUC Apr/May2010)
11. The marks obtained by a number of students in a certain subject are assumed to be
normally distributed with mean 65 and standard deviation 5. If 3 students are
selected at random from this group, what is the probability that atleast one of them
would have scored above 75? (AUC Apr/May2010)
(AUC Apr/May 2011)
12. The Probability distribution function of a random variable X is given by
,0 1
( ) (2 ) ,1 2
0 ,
, (0.2 1.2)
(0.5 1.5 / 1)
( ).
x x
f x k x x
k P x
P x X
f x
X
(i)Findthe valueof (ii)Find
(iii)What is
(iv)Findthedistribution functionof
otherwise (AUC Apr/May 2011)
MA 2261 Probability and Random process IV Sem ECE – S.SARULATHA Asst.Prof./MATHS
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13. Derive the mgf of Poisson distribution and hence or otherwise deduce its mean and
variance. (AUC Apr/May 2011)
14. Find the M.G.F of the random variable X having the probability density function
, 0( ) 4 2
0 ,
xe x
f x
elsewhere
x
. Also deduce the first four moments about the
origin. (AUC May/Jun 2012)
15. Given that X is distributed normally, if ( 45) 0.31 ( 64) 0.08P X and P X , find
the mean and standard deviation of the distribution. (AUC May/Jun 2012)
16. The time in hours required to repair a machine is exponentially distributed with
parameter 1
2
(1) What is the probability that the repair time exceeds 2 hours?
(2) What is the conditional probability that a repair takes atleast 10 hours
given that its duration exceeds 9 hours? (AUC May/Jun 2012)
17. If the probability density of X is given by 2(1 ) 0 1,