Problem set hso201

1. 1
Indian Institute of Technology Kanpur
Department of Humanities and Social Sciences
HSO201A Applied Probability and Statistics (2015-16-II)
Instructor: Professor Praveen Kulshreshtha
PRACTICE PROBLEM SET (PPS) # 3
Note:
 There are 8 questions in this problem set.
 As far as possible, complete answers should be given.
 ‘Quality of reasoning provided’ is very important.
 Make all necessary mathematical derivations to support your reasoning.
 Show all of your steps and work (including calculations) clearly.
1. Let X be a random variable with finite variance 2
. Show that for any real numbers
(constants) a and b, Var(aX + b) = a2
2
.
2. In 80% of all solar-heat installations, the utility bill is reduced by at least one-third.
Find the probability that the utility bill will be reduced by at least one-third in:
(a) five of six installations.
(b) at least five of six installations.
3. A shipment of 15 digital voice recorders contains 4 that are defective. If 5 of them are
randomly chosen for inspection, what is the probability that 2 of the 5 will be
defective?
4. Show that the Moment Generating Function (m.g.f.) for a random variable:
(a) X  B(n, p); n > 1 (integer) and 0 < p < 1 (Binomial Distribution), is given by:
MX(t) = [p + (1-p)]n
(b) X  Po();  > 0 (Poisson Distribution), is given by: MX(t) = 
(c) Y  NB(r, p); r  1 (integer) and 0 < p < 1, where Y = No. of failures before the rth
success occurs (Negative Binomial Distribution), is given by: MY(t) =
(d) Y  g(p); 0 < p < 1, where Y = No. of failures before the 1st
success occurs
(Geometric Distribution), is given by: MY(t) =
(e) X  Discrete Uniform(1, N); N > 1 (integer) (Discrete Uniform Distribution), is
given by: MX(t) =
(f) X  Uniform(a, b); a < b (Continuous Uniform Distribution), is given by:
MX(t) = , for t  0; and MX(0) = 1
(g) X  Gamma(, );  > 0 and  > 0 (Gamma Distribution), is given by:
MX(t) =

, for t < 
(h) X  2
(p); p > 0 (integer) (Chi Square Distribution), is given by:
MX(t) = , for t <
(i) X  exp();  > 0 (Exponential Distribution), is given by: MX(t) =

, for t < 

2. 2
5. Derive the Expectation (mean ) and Variance (2
) for a random variable X, where:
(a) X  B(n, p); n > 1 (integer) and 0 < p < 1 (Binomial Distribution)
(b) X  Po();  > 0 (Poisson Distribution)
(c) X  NB(r, p); r  1 (integer) and 0 < p < 1 (Negative Binomial Distribution)
(d) X  g(p); 0 < p < 1 (Geometric Distribution)
(e) X  Discrete Uniform(1, N); N > 1 (integer) (Discrete Uniform Distribution)
(f) X  Hypergeometric (N, n, K); N > 1 (integer), n (integer) < N; K (integer) < N
(Hypergeometric Distribution)
(g) X  Uniform(a, b); a < b (Continuous Uniform Distribution)
(h) X  Gamma(, );  > 0 and  > 0 (Gamma Distribution)
(i) X  2
(p); p > 0 (integer) (Chi Square Distribution)
(j) X  exp();  > 0 (Exponential Distribution)
6. A counter records an average of 1.3 gamma particles per millisecond coming from a
radioactive substance. Let X denote the random variable, which equals the count of
gamma particles during the next millisecond.
(a) Find a suitable probability distribution for X. What are the parameters of this
distribution?
(b) What are the mean and variance of the above distribution?
(c) Determine the probability that one or more gamma particles will be emitted during
the next millisecond.
7. In the city of Kanpur, the proportion of road sections requiring repairs in any given
year is a random variable having the beta distribution, with shape parameters  = 3
and  = 2.
(a) On average, what percentage of the road sections in Kanpur city require repairs in
any given year?
(b) Find the probability that at least half of the road sections in Kanpur city will
require repairs in any given year.
8. In the city of Kanpur, the daily consumption of electricity (in thousands of megawatt-
hours) can be treated as a random variable having the gamma distribution, with shape
parameter  = 2 and scale parameter  = 5. If the power plant of the city has a daily
capacity of 100 megawatt-hours, what is the probability that this power supply will be
inadequate on any given day?
ENJOY!