PROBABILITY & RANDOM PROCESSES
QUESTION BANK
UNIT – I
Part-A
1. What is a random experiment? Give an example. [April 2012]
2. Define a random variable with an example. [April 2011]
3. Define a discrete random variable with an example. [April 2013]
4. Define the probability mass function of a Discrete Random variable. NOV2015]
5. Define Cumulative Distribution function (cdf). [NOV 2014]
6. Define Probability distribution of a Random variable.
7. State the Properties of cdf F (x) .[April 2013]
[NOV2013]
8. A random variable X has the following Probability distribution.
X: -2 -1
P(x):
0.1
0
k 0.2
1
2
3
2k
0.3
3k . Find the value of k. [April 2017] [April 2015] [April 2016]
9. Define Bernoulli’s distribution.
[April 2017]
10. Define Binomial distribution.
11. Define Poisson distribution. [NOV2017]
12. Define Geometric distribution.
13. Define Hypergeometric distribution. [April 2016]
14. Define Negative Binomial Distribution
15. Define Discrete Uniform Distribution[NOV2017]
16. Define constant random variable
2
P ( X  1) and find P( X  0), P( X  3) . [April 2017]
3
4
18. If X is a Binomial distributed random variable with E(X)=2 and Var(X)= find P(X=5) [NOV 2018]
3
17. If X is a Poisson variate such that P ( X  2) 
19. Mean of a Binomial distribution is 20 and S.D is 4. Find the parameter of the
distribution?
[April 2018]

20. Find the mean and S.D of the distribution whose moment generating functions is 0.4et  0.6
[Nov2018]
1

21. Derive Probability Generating Function and Moment Generating Function of Binomial distribution.
[April 2019]
UNIT -II
1. Define a continuous Random variable with an example.
2. Define probability Density function.
3. State the Properties of Probability Density function.
4. State the relation between Probability distribution function and Probability
density function.
5. Check whether f ( x) 
1
1
,    x   is a probability density function.
 1 x2
6. Prove that f ( x)  6 x(1  x), 0  x  1 is a pdf.
7. If a continuous random variable X has a pdf f ( x)  kx2 , 0  x  3 then find the
value of k.
kx2 (1  x), 0  x  3
8. Find the value of k, so that f ( x)  
is a pdf of a continuous
0, otherwise
0, x  0
9. If a random variable X has a cumulative distribution function, F ( x)  
x
c(1  e ), x  0
Find p.d.f. f (x) and find the value of c.
0, x  1

10. If the c.d.f. F ( x)  c( x  1) 4 , 1  x  3 , then find P (1  X  2) .
1, if x  3

11. A continuous random variable X has a p.d.f. f ( x) 
1 2 x
x e , x  0.
2
Find mean and variance.
12. Define Uniform or Rectangular distribution
13. Define Exponential distribution.
14. State the memory less property of the Exponential distribution
15. Define Gamma distribution
16. State the Reproductive property of Gamma Distribution
2
17. Define Weibull Distribution.
18. Define Normal or Gaussian distribution
19. Define Standard Normal distribution
20. Show that the uniform distribution f ( x) 
is
1
, a  x  a, the mgf about the origin
2a
sinh at
at
21. If X is uniformly distributed random variable with mean 1 and variance 4 find
3
P ( X  0)
22. A random variable X has uniform distribution over (-3,3). Compute P( x  2)
23. Define the mathematical definition of Reliability.
24. State the properties of Reliability
25. Define Instantaneous failure rate
26. Define Cumulative hazard
27. Define conditional Reliability
27. Define Hypo exponential distribution.
28. Define Hyper exponential distribution.
29. Define Pareto or Double exponential distribution.
11 MARKS QUESTIONS
UNIT I -Discrete Random Variables
1. Define a Random variable. What are the characteristics of a Random variable?
[May 2008]
2. A random variable X has the following probability distribution.
X
0
P(X=x) a
1
2
3
4
5
6
3a
5a
7a
9a
11a 13a
Find a
3
7
8
15a 17a
Evaluate P(X < 3) and P(0 < X < 3)
Evaluate the mean & variance of X
Find the distribution function F(x).
[Nov2007,2014,April 2017]
2. If the random variable X takes the values 1,2,3,4 such that 2P[X=1] = 3 P[X=2] =
P[X=3]=5P[X=4]. Find the probability distribution and cumulative distribution function of X.
[Nov 2014,2015]
4. The number of telephone calls received in an office during lunch hour has the
probability function given below:
No. of callsX
0
1
2
3
4
5
6
P(X=x)
0.05
0.2
0.25
0.20
0.15
0.10
0.05
(i) Verify that it is really a probability function
(ii) Find the probability that there will be three or more calls
(iii) ) Find the probability that there will be an odd number of calls. [May 2008]
5. For a Binomial Distribution of mean 4 and variance 2 find the probability of getting
(i) atleast 2 successes (ii) at most 2 successes (iii) Find P(5  x  7)
[April 2015]
6. Out of 800 families, with 4 children each, how many families would you expect to have
(i) 2 boys & 2 girls (ii) atleast 1 boy (iii) atmost 2 girls (iv) children of both sex (v) no girls.
[Nov 2014]
7. The Probability function of an infinite series is given by P  X  j  
1
, j  1, 2,3,.. .
2j
(i). Verify that P(X) is really a probability mass function.(ii) . Find Mean and Variance
(iv) P  X  5
(iii) P[X is even]
(v) P[ X is divisible by 5]
[Apr 2014]
8. A Random variable x has the following probability distribution
X
-2
-1
0
1
2
3
P(X)
0.1
k
0.2
2k
0.3
3k
Find (i) k (ii) P(X<2) (iii) P(-2<X<2) (iv) Cdf (v) Mean
[Apr 2011,2013, Nov 2011,2012,2015]
4
9. The mistakes committed by a typist follow a Poisson distribution given below.
No. of mistakes per
page(x)
0
1
No. of pages (f)
142 156
2
3
4
5
69
27
5
1
Find the expected frequencies.
[Nov.2013]
10. The probability mass function of a RV, X is defined as P(X=0) = 3c2 , P(X=1) = 4c10c2, P(X=2) = 5c-1 for c >0 and P(X=r) = 0 , if r  0,1,2.
(i)
Find the value of c.
(ii)
P(0<x<2 / x > 0)
(iii) The distribution function of X
(iv)
The largest value of X for which F(x) < ½
(v)
The smallest value of X for which F(x)> ½
[May 2007, May 2014]
3
11. The probability mass function of a RV, X is defined as P(X= r) = kr , r =1,2,3,4
(i)
Find the value of k.
(ii)
P( 1/2 <x< 5/2 / x > 1)
(iii) The distribution function of X
(iv)
Find the mean and variance of X
12. A radioactive source emits particles at a rate of 10 per minutes in accordance with Poisson of 2/5 of being
recorded. Find the probability that atleast 4 particles are recorded in a 2-min period.
[April 2014]
13. The probability that a candidate can pass in an exam is 0.6.(i) What is the probability that he pass in the
third trial? (ii) What is the probability that he pass before the third trial? [April 2014]
14. A RV X has the following probability distribution
X
0
1
2
3
P(X=x)
0
k
2k
2k 3k

4
Find (i) the value of k (ii) P 1.5  x  4.5
Find also the distribution function of X.
5
6
7
k2
2k2
7 k 2 +k
x2
 (iii) The smallest value of  for which
[Nov2005,April ,Dec 2016]
15. Define the following distributions and find their mean variance, MGF and PGF.
a. Bernoulli’s distribution
b. Binomial distribution
5
Px    
1
.
2
c. Poisson distribution.
d. Geometric distribution.
e. Hyper geometric distribution.
f. Negative binomial distribution.
16. State and prove the memory less property of geometric distribution.
[April 2016]
17. P.T. Poisson distribution is a limiting case of Binomial distribution.
18. 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 on the fourth trial? Also find the probability that he
will pass the test in fewer than 4 trials.
[ Nov 2013]
19. Consider a plant manufacturing IC chips of which 10% are expected to be defective. The chips are packed
in packets containing 35 chips each. Determine how many packets are having more than 5 defectives out of
800 packets
[Nov2013]
20.The average number of traffic accidents on a certain section of high way is two per week. Assume that the
number of accidents follows a Poisson distribution. Find the probability of
(i) no accident in a week (ii) at most two accidents in a 2 week period.
[April 2015]
21 (a) For a Binomial distribution of mean 4 and variance 2 find the probalility of getting
(i) atleast 2 successes (ii) atmost 2 successes (iii) find p(5< x< 7)
[May 2017]
(b) The probability that a child exposed to certain contagious disease is 0.4. What will be the probability that
the tenth child exposed to the disease will be the third to catch it? [May 2017]
22. A taxi cab company has 12 Ambassadors and 8 Fiats. If 5 of these taxi cabs are in the worksho for repairs
and an Ambassador is as likely to be in for repairs as a Fiat,
(a) 3 of them are Ambassadors and 2 are Fiats?
(b) at least 3 of them are Ambassadors?
(c) all of the 5 are the same make?
[May 2017]
UNIT II - Continuous Random Variables
1. Define probability density function. What are the characteristics of a distribution function?
[Nov2007]
6
ax, (0,1)
a, (1,2)

2. If the pd of X is given by f(x) = 
(i) Find the value of ‘a (ii) Find the CDF of X. 3. For
3a  ax, (2,3)
0, elsewhere
0  x 1
x

the triangular distribution f ( x)  2  x 1  x  2 .Find the mean and variance and M.G.F
0
otherwise.

[April 2008, May 2014] 4.The time (hrs) required
1
to repair a machine is exponentially distributed with parameter  
2
(i) What is the probability that the repair time exceeds 2 hours? (ii) what is the probability that a repair takes
at least 10 hours given that its duration exceeds 8 hours? [April 2012,2015, May2014]
5. Find the mean and variance uniform distribution.
[Nov 2005,April 2014]
6. A continuous RV X has pdf f(x) = kx2 e  x , x  0 . Find k, mean and Variance of the RV X.
7. Define the following distributions also state the properties and find their mean and variance.
Exponential distribution. (Also find probability distribution function)
[May2007]
8. Gamma distribution (or) Erlang m – distribution (Also find probability distribution function)
[April 2016,Nov 2014]
Normal distribution
[Nov 06,May 2008]
9. Find the mean and variance of Weibull distribution (Also find probability distribution function)
[Nov2006]
10. Derive the relation between Poisson distribution and Gamma distribution.
11. If the life (in years) of a certain type of a car has a Weibull distribution with the parameter   2 , Find the
value of the parameter  , Given that probability that the life of the car exceeds five years is e 0..25 .For these
values of  ,  find the mean and variance of X.
[April 2007, 2015]
12. State and prove memory less property of exponential distribution.
[Nov2006]
13. Each of the 6 tubes of a ratio set has a life length (in years) which may be considered as a RV that follows
a Weibull distribution with parameters  = 25 and   2 . If these tubes function independently of one another,
What is the probability that no tube will have to be replaced during the first 2 months of service?
[ Nov 2013, April 2014]
14. For a certain normal distribution the first moment about 10 is 40 and that the fourth moment about 50 is 48.
What is mean and S.D. of the distribution.
15. An analog signal received at a defector (measured in microvolt) may be modeled as a Gaussian random
variable N(200,256) at a fixed point in time. What is the probability that the signal is larger than 240 micro
volts, given that it is larger than 210 microvolts?
[April 2012, Nov.2013]
7
16. In a certain city, the daily consumption of electric power in millions of kilowatt-hours can be treated as a
1
random variable having an general Gamma distribution with parameters   and k=3 . If the power plant of
2
this city has a daily capacity of 12 million kilowatt-hours, what is the probability that this power supply will be
inadequate on any given day?
[April 2012]
17. If the time T to failure a component follows Weibull distribution with parameter  and  , find the
instantaneous failure rate at time t of the component.
[April 2012]
18. Trains arrive at station at 15 minutes interval, starting at 4 a.m. If a passenger arrives at the station at time t
that is uniformly distributed between 9.00 and 9.30 am., find the probability that he has to wait for the train for
(i) less than 6 minutes (ii) more than 10 minutes.
[Nov 2013]
19. The density function of the time to failure in years of an item manufactured by a certain company is
200
f (t ) 
, t  0 , (i) Derive the reliability function (ii) Determine the reliability for the first year of
(t  10) 3
operation (iii) What is the design life for reliability 0.95?
20. What is the variance of the variable which is uniformly distributed?
[Nov 2013]
[May 2014]
21. Let x be the service life of a semi conductor having Weibull distribution with  =0.025 and  = 0.5 as
parameters. Find the probability that the semiconductor will be working after 3000 hours.
[April 2015]
22. A certain type of storage battery lasts on the average 3.0 years with standard deviation of 0.5 year.
Assuming that the battery lives are normally distributed, find the probability that a given battery will last less
than 2.3 years.
23. A random variable x has uniform distribution over (-3,3). Compute (i) p(x<2) (ii) p(1<x<2) (iii) Find k for
p(x>k)=1/3.
[April 2015]
24. The millage which car owners get with a certain kind of radial tyre is a random variable having an
exponential distribution with mean 40000 km. Find the probabilities that one of these tyres will last (i) at least
20000 km and (ii) atmost 30000 km.
[ April 2017]
(b) Derive the mean and variance of Gamma distribution.
[April 2017]
25. If the life X (in years) of a certain type of car has a Weibull distribution with parameter  =2. Find the
value of the parameter , given that probability that the life of the car exceeds 5 years is . For these values of 
and  , find the mean and variance.
[April 2017]
8