probability distribution

Probability Distribution

One Mark Questions

PREPARED BY:R.RAJENDRAN. M.A., M. Sc., M. Ed.,K.C.SANKARALINGA NADAR HR. SEC. SCHOOL,CHENNAI-21

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

(a) 1/3 (b) 1/6

(c) 1/9 (d) 1/12

2. If f(x) = – < x < is a p.d.f of a continuous

random variable X, then the value of A is

(a) 16 (b) 8

(c) 4 (d) 1

216

1

x

A


3. A random variable X has the following probability

distribution

X 0 1 2 3 4 5

P(X = x) ¼ 2a 3a 4a 5a ¼

then P(1 X 4)

(a) 10/21 (b) 2/7

(c) 1/14 (d) 1/2


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


5. A random variable X has the following p.d.f

then the value of is

(a) (b)

(c) (d) –1 or

X 0 1 2 3 4 5 6 7

P(X = x) 0 k 2k 2k 3k k2 2 k2 7k2+k

8

110

1

10

10


6. X is a discrete random variable which takes 0, 1, 2 and

P(X = 0) = P(X = 1) = then the value of P(X

= 2)is

(a) (b)

(c) (d)

169

144

169

1

169

145

169

2

169

24

169

143

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


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


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

(a) (b)

(c) (d)

3

55

3

9

5

5

9


13. The mean of a binomial distribution is 5 and its

standard deviation is 2. Then the value of n and p is

(a) (b)

(c) (d)

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

25,5

4

25,5

1

5

4,25

5

1,25


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


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

(a) (b) (c) (d) 20

1

125

18

25

4

10

3


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

(a) (b) (c) (d) 2

1

51

26

51

25

102

25


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


23. If f(x) is a p.d.f of a normal distribution with mean µ

then is

(a) 1 (b) 0.5 (c) 0 (d) 0.25

24. The random variable X follows normal distribution

f(x) = then the value of c is

(a) (b) (c) (d)

dxxf )(

25

)100(2/1 2 x

ce

2 2

125 25

1


25. If f(x) is a p.d.f of a normal variate X and X – N(, 2) then

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

dxxf )(


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 =


29. In the following probability distribution function

the value of ‘a’ is

(a) (b) (c) (d) 1

30. If 3 dice are thrown once, the number of elements in the

sample space is

(a) 6 (b) 36 (c) 216 (d) none

X 1 2 3 4 5 6P(X) a 2a 3a 4a 5a 6a

6

15

1

21

1


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)

(a) (b) (c) (d) 36

2536

10

36

1

36

6


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


35. A continuous random variable X follows the probability

law, f (x) = where A = , then

the value of k is

(a) A (b) 1/A

(c) A2 (d) 0

elsewhere

xxkx

,0

10,)1( 10

1

0

10)1( dxxx


36. If the probability law density function of a random

variable is given by f (x) = , then

the value of k is

(a) 2/3 (b) 3/2

(c) 2 (d) 3

elsewhere

xxk

,0

10,)1( 2


37. F (x) = , – < x < is a distribution

function of a continuous variable X, then P(x 1) is

(a) 3/4 (b) /4

(c) ½ (d) ¼

x1tan

2

1


38. If f (x) = 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

elsewhere

exx

A

,0

1, 3


39. The probability density function of a random variable X

is f(x) = then the value of k is

(a) / (b)

(c) 1/ (d) /

elsewhere

x

,0

0,,,ekxα x β1α


40. For the distribution function given by

F(x) = , then p.d.f f(x) is

(a) (b) x; x > 0

(c) cannot be determined (d) 2x; x<0

1,1

10,

0,02

x

xx

x

elsewhere

xx

,0

10,2


41. A continuous random variable x has the p.d.f defined by

f(x) = , then c is

(a) 1 (b) a

(c) 1/a (d) – a

elsewhere

xce ax

,0

0,


42. A random variable X has a probability density function

f(x) = , then k is

(a) 2 (b)

(c) (d)

elsewhere

xk

,0

20,

2

1

1


43. The mean of the distribution f(x) = is

(a) 1/3 (b) 3

(c) – 1/3 (d) – 3

44. The mean value of p.d.f f(x) = is

(a) 0 (b) 24

(c) 1/12 (d) 12

elsewhere

xe x

,0

0,3 3

elsewhere

x

,0

1212,24

1


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


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


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


50. For a standard normal distribution the mean and

variance are

(a) , 2 (b) ,

(c) 0, 1 (d) 1, 1


51. The p.d.f of the standard normal variance Z is (z) =

(a) (b)

(c) (d)

2

2

1

2

1 ze

2

2

1 ze

2

2

1

2

1 ze

2

2

1

2

1 ze


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)


53. The probability density function of the normal

distribution is f(x) = – < x < , then

the mean =

(a) 1 (b) 2

(c) – 2 (d) 4

xxKe 42 2


54. For the p.d.f of the normal distribution function

– < x < , then the mean =

(a) 3 (b) 2/3

(c) 3/2 (d) 6

xxCe 32


55. The p.d.f of the normal distribution is

– < x < , then the variance 2 =

(a) ¼ (b) 1/2

(c) 1 (d) 2

xxKe 42 2


56. The p.d.f of the normal distribution is

– < x < , then the variance 2 =

(a) ½ (b) 2

(c) 3/2 (d) 1/2

xxCe 32


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)


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


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


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

probability distribution

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