Applied Probability and Statistics Unit I:Random Variables
Applied Probability and Statistics Unit I:Random Variables
Applied Probability and Statistics Unit I:Random Variables
1. Define Poisson distribution and State any two instances where Poisson
distribution may be successfully employed.
x , 0 x 1
2. For the triangular distribution f ( x) 2 x , 1 x 2
0 , otherwise
Find the moment generating function.
3. Find the mean and variance of the distribution whose moment generating
function is (0.4e t +0.6) 2
x e x , x 0
4. A random variable X has density function given by f ( x )
0, x 0
Find the moment generating function.
5. Find the mean of the Poisson distribution.
2x , 0 x 1
7. (a) Let f ( x )
0 , otherwise
Find probability density function of Y=4X+2.
1 , x 1
If X is a continuous random variable with F x k x 1 ,1 x 3
4
(b)
0 , x3
Find the value of k, probability density function and P(X <2).
0 , x3
Find the value of k.
2. X and Y are independent random variables with variance 2 and 3. Find
the variance of 3X+4Y.
3. Define moment generating function.
4. Write down the properties of moment generating function.
5. If X is a continuous random variable with
1, x 1
F x k x 1 ,1 x 3
4
0, x 3
Find P(X<2) if the value of k= 1/16.
(b) If X is a uniform random variable in the interval (-2,2). Find the pdf
of Y=X 2 .
ì 1 -x
ï e 2 ;x >0
1. Let the random variable X has the pdf f ( x) = í 2
ï
îï 0 ; otherwise
Find the mean.
2. If X is uniformly distributed over (-1,1) find the density function of
.
3. The first four moments of a distribution about X=4 are 1,4,10 and 45
respectively. Show that the mean is 5, variance is 3, μ 3 =0 and μ 4 =26.
10. (a) Find the moment generating function of the random variable X
ì x -x
ï e 2 , x >0
having the pdf f ( x) = í 4 also deduct the first four
ï
ïî 0 , elsewhere
moment about the origin.
12. (a) Find the moment generating function of the binomial random
variable with parameters m and p and hence find its mean and
variance.
ì -x
ï 2 , x³ 0
(b) If P( x ) = í xe
ï 0 , x <0
î
i. Show that P(x) is a pdf.
ii. Find F(x)
(b) State and prove memory less property of geometric distribution. Also
derive mean and variance of a G.D.
15. (a) A die is tossed until 6 appears. What is the probability that it must
be tossed more than 4 times.
(b) 6 dice are thrown 729 times. How many times do you expect at least
three dice to show 5 or 6?
16. (a) A coin is tossed until the first head occurs. Assuming that the tosses
are independent and the probability of a head occurring is p, find the
value of p, so that the probability that an odd number of tosses is
required is equal to 0.6. Can you find a value of p so that the
probability is 0.5 that an odd number of tosses are required?
(b) Find the mean, variance and moment generating function of a
random variable uniformly distributed in the interval (a, b).
(b) The mileage which car owners get with a certain kind of radial tire is
a random variable having an exponential distribution with mean
40,000 km. find the probabilities that one of these tires will last
(i) at least20,000 km
(ii) at least 30,000 km.
1. Suppose that a bus arrives at a station every day between 10.00 am and
10.30 am at random. Let X is the arrival time. Find the distribution
function of X and sketch its graph.
2. If X is a random variable uniformly distributed in (0,1), find the pdf of
Y = sin X.
3. Define the Uniform Distribution function.
4. What is the pdf of an exponential distribution?
5. The mileage which car owners get with a certain kind of radial tire is a
random variable having an exponential distribution with mean 40,000
km. Find the probabilities that one of these tires will last atleast 20,000
km.
18. (a) The mean yield for one-acre plot is 662 kilos with a S.D. 32 kilos,
Assuming normal distribution, how many one-acre plots in a batch of
1,000 plots would you expect to have yield
(i) over 700 kilos
(ii) below 650 kilos
(iii) what is the lowest yield of the best 100 plots?
(b) Find the M.G.F, Mean and Variance for Normal distribution.
19. (a) The time (in Hours) required to repair a machine is exponentially
distributed with parameter λ=½. What is the probability that the
repair time exceeds 2 h? What is the conditional probability that a
repair takes at least 10 h given that its duration exceeds 9h?
(b) The local authorities in a certain city install 10,000 electric lamps in
the streets of the city. If these lamps have an average life of 1000
burning hours with a S.D. of 200 hours, assuming normality, what
number of lamps might be expected to fail
(i) in the first 800 burning hours
(ii) between 800 and 1200 burning hours
20. (a) Find the m.g.f, mean and variance of exponential distribution.
(b) Prove that the Poisson distribution as the limiting case of binomial
distribution.