Download MCA-I Semester Regular Examinations, 2013

MCA-I Semester Regular Examinations, 2013-2014
Probability and statistical Applications
UNIT – II RANDOM VARIALES
1(a) A random variable X has the following probability function
X
1
2
3
4
P(X)
k
3k
5k
7k
(i) Find the value of k. (ii) P(1<X5) (iii) Mean (iv) Variance.
5
9k
6
11k
(b) the probability density function of a random variable X is given as f ( x)  e  x for X  0
= 0 otherwise. Then find Mean
[6+6]
and Variance of X.
(OR)
( c) Derive the mean and variance of Binomial distribution.
[5+7]
(d) the distribution of typing mistakes committed by a typist is given below. Assuming the distribution to be
poisson, find the expected frequencies.
X
f
0
125
1
95
2
49
3
20
4
8
5
3
2(a) Find the expected value of X and standard deviation for the following discrete distribution
X
8
12
16
20
24
f
1/8
1/6
3/8
1/4
1/12
(b) The diameter of an electric cable is assumed to be continuous random variable with probability density
function f ( x)  6 x(1  x)0  x  1 justify. Also find the mean and variance of the distribution. [6+6]
(OR)
( c) Ten unbiased coins are tossed simultaneously. Find the probability of obtaining 9i) exactly 6 heads (ii) at
least 8 heads (iii) no heads (iv) at least one head (v) not more than three heads (vi) at least 4 heads.
(d) set a binomial distribution to the following data:
X
0
1
2
f
28
62
46
[6+6]
3
10
4
4
3(a) It is claimed that a random sample of 100 tyres with a mean life of 15269 kms is drawn from a
population of tyres which has a mean life of 15200 kms and a S.D. of 1248 kms. Test the validity of the
claim at (i) 55 and (ii) 1% level of significance.
(b) A weighing machine without any display was used by an average of 320 persons day with a standard
deviation of 50 persons when an attractive display was used on the machine, the average for 100 days
increased by 15 persons. Can we say that the display did not help much? Use level of significance of
0.05.
[6+6]
(OR)
( c) Define random variables, continuous random variables, and distributive function, cumulative distributive
Function. A random variable X has the following probability function
X=x
0
1
2
3
4
5
6
7
2
P(X=x)
0
k
2k
2k
3k
7k 2 +k
2 k2
k
(i) find the value of k (ii) P(X5), P(X>5), P(X<6), P(X6) (iii) P(0<X<6), P(0<X<5)
(d) The probability that a bomb dropped from a plane will strike the target is 1/5. If six bombs are dropped
find the probability that (i) exactly two will strike target, (ii) at least two will strike the target.
[6+6]
4(a) Define mean and variance of normal distribution, population, sample, parameter, sampling distribution
and standard error.
(b) From a lot of items containing 3 defectives a sample of 4 items is drawn at random. Let the random
variable X denote the number of defectives items in the sample. Find the probability distribution of X
when the sample is drawn without replacement.
[6+6]
(OR)
(c ) Find the binomial distribution to the following data
X
0
1
2
3
4
5
f
2
14
20
34
22
8
(d) Suppose 3% of bolts made by a machine are defective, the defects occurring at random during [6+6]
production. If bolts are packed 50 per box find the probability that a given box will contain 5 defectives?
5(a) A random sample of size 100 is taken from an infinite population having the mean  = 78 and the
variance 2 = 256. What is the probability that x will be between 75 and 78.
(b) The guaranteed average life of a certain type of electric bulbs is 1500 hrs. with a s.d. of 120 hrs. It is
decided to sample the output so as to ensure that 95% of bulbs do not fall short of the guaranteed average
by more than 2%. What will be the minimum sample size.
[6+6]