Biplab Sinha Mahapatra
School of Applied Science and Humanities
Haldia Institute of Technology
Mathematics-III (Probability & Statistics)
Module-I
Probability Distribution
1. A random variable X has the density function
{
Find the mean and variance of X.
Ans:
| |
{
2. Show that the function
is a probability density function and hence find the
corresponding distribution function.
Ans:
{
3. For what values of a, P(x) will be a pmf. of a discrete random variable X, whose probability distribution is given below:
x
0 1
2
3 4
2
P(x): a 2a 7a 2a a
Hence find (i)
.
Ans:
(i) (ii)
[
] and 0
4. Let X is a random variable whose density function f forms an isosceles triangle above the unit interval
elsewhere. Then prove that
a) Height of the triangle is 2.
b) Formula for pdf is
{
c) The mean of X is .
5. The diameter of an electric cable, say X is assumed to be a continuous random variable with pdf
i) Check that
is a pdf
ii) Determine a number b such that
.
Ans: (ii)
.
6. A random variable X has the following probability mass function:
1 0
1
2
3
0.1 k
0.2 2k 0.3 3k
i) Find k
ii) Evaluate
iii) Determine the distribution function
Answer: i)
.
.
ii) 0.5, 0.8, 0.4
iii)
{
7. The pdf of a random variable X is
i)
ii)
, find
{
,
|
(|
)
Answer: i) ii)
8. The pdf of a random variable X is given by
Compute Mean and Variance.
Answer: 1,
9. Let
Find k such that
is a density function. Find also the corresponding
distribution function.
Answer:
10. For a random variable X,
, find
.
Answer: 4
11. The length of life of tyre manufactured by a company follows a continuous distribution given by the density
function
. Find k and find the probability that a randomly selected tyre would
{
function for at least 1200 hours.
Answer:
0.45
12. The pdf of a continuous random variable X is given by
Variance of X. Given
. Find the Expectation and
{
.
Answer:
13. Find the value of the constant k such that
). Also find
(
Ans: (i)
.
, (ii) 0.5, (iii) 0.5
14. The distribution function of a random variable X is
(i)
(ii)
(iii)
is a possible density function and compute
{
The constant c
The density function.
{
. If
find
(iv)
(v)
Ans: (i) 1/27 (ii)
(iii) 26/27 (iv) 7/27 (v) 8/27
15. The pdf of a random variable X is
(i)
C
(ii)
(iii)
(iv)
(v)
(vi)
(
. Find
)
Mean and S. D.
Ans: (i) 3, 1/8, 37/64, 13/32, 3/5
16. The pdf of a R. V. X is given by
(i) The value of k
(ii) The distribution function
(iii)
(
. Determine
)
(iv) Mean and Variance
(v)
(vi)
Ans: (i) 6 (ii)
(iii) 11/32 (iv) 3/2, 1/20 (v)
(vi)