Transcript 9.PROBABILITYx

One Mark Questions
Probability Distribution
Choose the Correct Answer
1. If f(x) = kx2, 0 < x < 3
0, elsewhere
is a probability density
function then the value of k is
2.
(a) 1/3
(b) 1/6
(c) 1/9
(d) 1/12
A
1
If f(x) =
– < x <  is a p.d.f of a continuous
2
 16  x
random variable X, then the value of A is
(a) 16
(b) 8
(c) 4
(d) 1
Choose the Correct Answer
3. A random variable X has the following probability
distribution
X
0
P(X = x) ¼
1
2
3
4
5
2a
3a
4a
5a
¼
then P(1  X  4)
(a) 10/21
(b) 2/7
(c) 1/14
(d) 1/2
Choose the Correct Answer
4. A random variable X has the following probability
mass function as follows:
X
-2
3
1
P(X = x)
/6
/4
/12
then the value of  is
(a) 1
(b) 2
(c) 3
(d) 4
Choose the Correct Answer
5. A random variable X has the following p.d.f
then the value of  is
X
0
1
2
3
4
5
6
P(X = x) 0
k
2k
2k
3k
k2
2 k2 7k2+k
(a) 1
8
1
(b)
10
(c)
1
(d) –1 or
10
0
7
Choose the Correct Answer
6. X is a discrete random variable which takes 0, 1, 2 and
144
1
P(X = 0) =
P(X = 1) =
then the value of
169
169
P(X = 2)is
145
(a)
169
24
(b)
169
2
(c)
169
143
(d)
169
Choose The Correct Answer
7. Given E(X + c) = 8 and E(X – c) = 12 then the value of
c is
(a) – 2
(b) 4
(c) – 4
(d) 2
8. X is a random variable taking the value 3, 4 and 7 with
probabilities 1/3, ¼ and 4/7 then E(X) is
(a) 5
(b) 6
(c) 7
(d) 3
Choose The Correct Answer
9. Variance of the random variable X is 4. Its mean is 2,
then E(X2) is
(a) 2
(b) 4
(c) 6
(d) 8
10. 2 = 20, 2’ = 276 for a discrete random variable X,
then the mean of the random variable X is
(a) 16
(b) 5
(c) 2
(d) 1
Choose The Correct Answer
11. Var (4X + 3) is
(a) 7
(b) 16 Var(X)
(c) 19
(d) 0
12. In 5 throws of a die, getting 1 or 2 is a success. The
mean number of successes is
3
5
(a)
(b)
5
3
5
(c)
9
9
(d)
5
Choose The Correct Answer
13. The mean of a binomial distribution is 5 and its
standard deviation is 2. Then the value of n and p is
(a)  4 ,25 
(b)  25, 4 
1 
 ,25 
5 
 1
 25, 
5

5
(c)


(d)
5
14. If in a Poisson distribution P(X = 0) = k, then the
variance is
(a) log(1/k)
(b) log k
(c) e
(d) 1/k
Choose The Correct Answer
15. If the mean and standard deviation of a binomial
distribution are 12 and 2 respectively. Then the value
of its parameter p is
(a) ½
(b) 1/3
(c) 2/3
(d) ¼
16. If a random variable X follows Poisson distribution
such that E(X2) = 30 then the variance of the
distribution is
(a) 6
(b) 5
(c) 30
(d) 25
Choose The Correct Answer
17. In 16 throws of a die getting an even number is
considered a success. Then the variance of the
successes is
(a) 4
(b) 6
(c) 2
(d) 256
18. A box contains 6 red and 4white balls. If 3 balls are
drawn at random, the probability of getting 2 white
balls is
1
(a)
20
18
(b)
125
4
(c)
25
3
(d)
10
Choose The Correct Answer
19. For a Poisson distribution with parameter  = 0.25 the
value of the 2nd moment about the origin is
(a) 0.25
(b) 0.3125 (c) 0.0625
(d) 0.025
20. If 2 cards are drawn from a well shuffled pack of 52
cards, the probability that they are of the same colours is
1
(a)
2
26
(b)
51
25
(c)
51
25
(d)
102
Choose The Correct Answer
21. In a Poisson distribution if P(X = 2) = P(X = 3) then the
value of its parameter  is
(a) 6
(b) 2
(c) 3
(d) 0
22. The distribution function F(X) of a random variable X is
(a) a decreasing function
(b) an increasing function
(c) a constant function
(d) increasing first and then decreasing
Choose The Correct Answer
23. If f(x) is a p.d.f of a normal distribution with mean µ
then  f ( x)dx is

(a) 1
(b) 0.5
(c) 0
(d) 0.25
24. The random variable X follows normal distribution
f(x) =
ce
(a) 2
1 / 2 ( x 100 ) 2
25
then the value of c is
1
(b)
2
(c) 5 2
1
(d)
5 2
Choose The Correct Answer
25. If f(x) is a p.d.f of a normal variate X and X – N(, 2) then

f ( x)dx is


(a) undefined
(b) 1
(c) 0.5
(d) – 0.5
26. The marks obtained by 400 students in a mathematics test were
normally distributed with mean 65. If 120 students got more
marks above 85, the number of students securing marks between
45 and 65 is
(a) 120
(b) 20
(c) 80
(d) 160
Choose The Correct Answer
27. In a distribution of X, the mean and variance are 1.5 and
0.5, then E(X2) is
(a) 1.75
(b) 1.5
(c) 2.75
(d) 2.25
28. The normal probability curve is symmetrical about
(a) the x-axis
(b) the y-axis
(c) the line x = µ
(d) the line x = 
Choose The Correct Answer
29. In the following probability distribution function
X
1
2
P(X) a
2a
the value of ‘a’ is
3
3a
4
4a
5
5a
6
6a
1
1
(a)
(b)
(c) 1
(d) 1
5
6
21
30. If 3 dice are thrown once, the number of elements in the
sample space is
(a) 6
(b) 36
(c) 216
(d) none
Choose The Correct Answer
31. If X is a continuous random variable then P(a < X < b) =
(a) P(a  X  b)
(b) P(a < X  b)
(c) P(a  X < b)
(d) all the three
32. Two dice are rolled and X be the random variable of
getting number of ‘3’s. Then P(X = 2)
25
(a)
36
10
(b)
36
1
(c)
36
(d)
6
36
Choose The Correct Answer
33. A continuous random variable X has probability density
function f(x) then
(a) 0  f(x)  1
(b) f(x)  0
(c) f(x)  1
(d) 0 < f(x) < 1
34. A discrete random variable X has probability mass
function p(x) then
(a) 0  p(x)  1
(b) p(x)  0
(c) p(x)  1
(d) 0 < p(x) < 1
Choose The Correct Answer
35. A continuous random variable X follows the probability
kx(1  x)10 ,0  x  1
law, f (x) = 
0, elsewhere
then the value of k is
(a) A
(b) 1/A
(c) A2
(d) 0
1
where A =  x(1  x)10 dx ,
0
Choose The Correct Answer
36. If the probability law density function of a random
k (1  x) 2 ,0  x  1
variable is given by f (x) = 
0, elsewhere
value of k is
(a) 2/3
(b) 3/2
(c) 2
(d) 3
, then the
Choose The Correct Answer
37. F (x) = 1    tan 1 x  , –  < x <  is a distribution
2

function of a continuous variable X, then P(x  1) is
(a) 3/4
(b) /4
(c) ½
(d) ¼
Choose The Correct Answer
38. If f (x) =
A
3
 ,1  x  e
x
0, elsewhere
is a probability density
function of a continuous random variable X, then the
value of A is
(a) 1
(b) 3
(c) 1/3
(d) log3
Choose The Correct Answer
39. The probability density function of a random variable X
kx α 1e β x , x,  ,   0
is f(x) = 
then the value of k is
0, elsewhere
α
(a)  / 
(b) 
(c) 1/ 
(d) /
Choose The Correct Answer
40. For the distribution function given by
F(x) =
0, x  0
 2
 x ,0  x  1
1, x  1

, then p.d.f f(x) is
2 x,0  x  1
(a) 
(b) x; x > 0
(c) cannot be determined
(d) 2x; x<0
0, elsewhere
Choose The Correct Answer
41. A continuous random variable x has the p.d.f defined by
f(x) =
ce  ax ,0  x  

0, elsewhere
, then c is
(a) 1
(b) a
(c) 1/a
(d) – a
Choose The Correct Answer
42. A random variable X has a probability density function
f(x) =
k ,0  x  2
, then k is

0, elsewhere
(a) 2
(b) 1
1
(c)
2
(d) 

Choose The Correct Answer
3e 3 x ,0  x  
43. The mean of the distribution f(x) = 
0, elsewhere
(a) 1/3
(b) 3
(c) – 1/3
(d) – 3
44. The mean value of p.d.f f(x) =
1
 ,12  x  12
 24
0, elsewhere
(a) 0
(b) 24
(c) 1/12
(d) 12
is
is
Choose The Correct Answer
45. Mean and variance of binomial distribution are
(a) nq, npq
(b) np, npq
(c) np, np
(d) np, npq
46. In a binomial distribution if n = 5, P(X = 3) = 2P(X = 2)
then
(a) p = 2q
(b) 2p = q
(c) p = q
(d) 3p = 2q
Choose The Correct Answer
47. For a Binomial distribution with mean 2 and variance
4/3, p = ?
(a) 2/3
(b) 1/3
(c) ¾
(d) 2/3
48. In a Poisson distribution if P(X = 2) = P(X = 3), then
mean  = ?
(a) 1
(b) 2
(c) 3
(d) 4
Choose The Correct Answer
49. Which of the following is or are correct regarding
normal distribution curve?
(1) Symmetrical about the line X =  mean
(2) Mean = median = mode
(3) Unimodal
(4) Point of inflexion are at X =   
(a) 1, 2 only
(b) 2, 4 only
(c) 1, 2, 3 only
(d) all
Choose The Correct Answer
50. For a standard normal distribution the mean and
variance are
(a) , 2
(b) , 
(c) 0, 1
(d) 1, 1
Choose The Correct Answer
51. The p.d.f of the standard normal variance Z is (z) =
(a)
(c)
1
2 
1
2
e
e
1
 z2
2
1 2
z
2
(b)
1 z2
e
2
(d)
1
2
e
1
 z2
2
Choose The Correct Answer
52. If X is normally distributed with mean 6 and standard
deviation 5 and Z is the corresponding normal variate,
then P(0  x  8) =
(a) P(– 1.2 < z < .04)
(b) P(– 0.12 < z < 0.4)
(c) P(– 1.2 < z < 0.4)
(d) P(– 0.12 < z < 0.04)
Choose The Correct Answer
53. The probability density function of the normal
distribution is f(x) = Ke
2 x 2  4 x
–  < x < , then the
mean  =
(a) 1
(b) 2
(c) – 2
(d) 4
Choose The Correct Answer
54. For the p.d.f of the normal distribution function Ce  x
–  < x < , then the mean  =
(a) 3
(b) 2/3
(c) 3/2
(d) 6
2
3 x
Choose The Correct Answer
55. The p.d.f of the normal distribution is Ke
–  < x < , then the variance 2 =
(a) ¼
(b) 1/2
(c) 1
(d) 2
2 x 2  4 x
Choose The Correct Answer
56. The p.d.f of the normal distribution is Ce
–  < x < , then the variance 2 =
(a) ½
(b) 2
(c) 3/2
(d) 1/2
 x 2 3 x
Choose The Correct Answer
57. For a normal distribution with mean  = 34 and variance
2 = 16, then P(30 < x < 60) = (where X is normal
variate and Z is the corresponding standard normal
variate)
(a) P(–0.25 < z < 0.25)
(b) P(0 < z < 1.625)
(c) P(0.25 < z < 1.625)
(d) P(–0.25 < z < 1.625)
Choose The Correct Answer
58. A random variable X has the following probability mass
function, then k is
x
0
1
2
3
4
5
6
P(x)
k
3k
5k
7k
9k
11k
13k
(a) 1/13
(b) 1/36
(c) 1/49
(d) 1/7
Choose The Correct Answer
59. A discrete random variable takes
(a) only a finite number of values
(b) all possible values between certain given limits
(c) infinite number of values
(d) a finite or uncountable number of values
Choose The Correct Answer
60. A continuous random variable takes
(a) only a finite number of values
(b) all possible values between certain given limits
(c) infinite number of values
(d) a finite or uncountable number of values