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UNIVERSITY OF CALICUT

SCHOOL OF DISTANCE EDUCATION B Sc. Mathematics (2011 Admission Onwards) II SEMESTER Complementary Course PROBABILITY DISTRIBUTIONS (STATISTICS) QUESTION BANK 1. The joint cumulative distribution function F(x,y) lies within the values a). -1 and +1 c). -∞ and 0 b). -1 and 0 d). 0 and 1 2. If x and y are two independent random variables then f(x,y) = .... a). f(x)+f(y) c). f(x).f(y) b). f(x)-f(y) d). f(x)/f(y) 3. The value of F (-∞+ ∞) = .... a). 0 b). 1 c).+∞ d). -∞ 4. If X and Y are two independent r.v.’s the cumulative distribution function F(x,y) is equal to a).F1(x).F2(y) b).P(X ≤ x , Y ≤ y ) c).both a and b d).neither a nor b 5. If X and Y are two independent r.v’s then a).E(XY)=1 b).E(XY) = 0 c).E(XY)=E(X).E(Y) d).E(XY) = a constant 6. If X and Y are two random variables such that their expectations exist and P(x ≤ y)=1 then a).E(X) ≤ E(Y) c).E(X)=E(Y) b). E(X) ≥ E(Y) d).None of the above 7. If X is a random variable then E(X-E(X))2 = ….. a).µ1 b) .µ2 c) .µ3 d).µ4 8. If X and Y are two random variables then a).E(XY)2 = E(X2).E(Y2) b). E(XY)2 = E(X2Y2) c). E(XY)2 ≥ E(X2)E(Y2) d). E(XY)2 ≤ E(X2)E(Y2) 9. If X and Y are two random variables then the expressionE(X-E(X ))(Y- E(Y )) is called: a).V(X) c).Cov(X,Y) b).V(Y) d).Correlation of X and Y 10. If X is a random variable with mean then E(X- )r is called : a).variance c) .rth central moment th d). none of the above b).r raw moment Probability Distributions 

 

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11. If X is a r.v having pdf f (x),then E(X) is called ……. a).Arithmetic mean b).Geometric mean c).Harmonic mean d)First Quartile 12. If X is a r.v. and r is an integer, then E (Xr) represents a) .rth central moment b).rth raw moment c) .rth factorial moment d). none of these 13. For Bernoulli distribution with probability p of a success and q of a failure, the relation between mean and variance that holds is a).mean p b).q = p d).q ≠ p 42. Let X follows a poisson distribution with parameter , then mgf of X is c).  a). b). d).  43. The distribution function of a continuous uniform distribution of a variable X lying in the interval (a,b) is c).  a). b).  d).  44. If X ~P (1) and Y ~P (2), then the probability P (X + Y < 3) is a). c).4 b).3 d).8.5 45. Which of the following real life situations follow Poisson distribution a).The number of printing mistakes per page of a book b).The number of defects per item produced c).The number of persons arriving in a queue d).All the above. Probability Distributions 

 

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46. Normal distribution was invented by a).Laplace c).Gauss b).De-Moivre d).All the above. 47. If and are two independent Poisson variates with parameters and respectively, then the variable ( + follows. a).Binomial distribution with parameters ( + ) b).Poisson distribution with parameter ( + ) c).Either of a and b d).Neither of a and b 48. The skewness of a binomial distribution will be zero if c).p = a).p < b).p > d).p < q 49. Binomial distribution tends to Poisson distribution when a).n→∞ , p →0 and np = λ(finite) b). n→∞ , p → and λ = λ(finite) c). n→0 , p → 0 and np → 0 d). n→0 , p → 0 and np → -∞ p; p +q = 1; x = 2, 3, 4, .... 50. A discrete random variable has pmf p(x) = The value of k is c). a). b).

d).

51. If a discrete random variable takes on four values -1,0,3,4 with probabilities 1/6,k,1/4 and 1-6k, where k is a constant. The value of k is a).1/3 c).1/12 b).2/9 d).5/24 52. Let X be a continuous random variable with probability density function is f (x) = kx; 0 ≤ x ≤ 1 = k; 1 ≤ x ≤ 2 = 0; otherwise. The value of k is equal to a).1/4 c).2/5 b).2/3 d).5/4 53. For the distribution function of a random variable X, F (5) F (2) is equal to a).P (2 < x < 5) c).P (2 ≤ x ≤ 5) b).P (2 ≤ x < 5) d). P (2 0 d). e , x > 0 Probability Distributions 

 

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99. If Var(x) = 1, then Var(2X 3)is a).5 b).13 c).4 d).2 is minimum when 100. E a).k < E(X) c).k = E (X ) b).k > E(X ) d).k = 0 101. Let X and Y be two bivariate continuous random variables and f(x,y) be its probability density function. Then random variables x and y are independent if f(x,y) = f(x). f(y). Here f(x) denote a) Values of function ‘f’ at ‘x’ b)

c) 

f x, y dy

f x, y dx f x, y dx dy

d)

102. Two fair dice are tossed. Let x denote the difference (absolute) of 2 face numbers and y their sum. Then P(x=2, y=7) is a) c) 0 b) d) 103. Two perfect dice are thrown simultaneously, If x = face number of the first dice and y that on the 2nd dice, which of the following is zero? a) P(x=1, y=1) c) P (x=2, y>3) b) P (x+y =1) d) P (x=y) 104. Following table gives the joint probability distribution of (x, y)

x

1

y 1

1

2

2

3

3

P(x)

6

30 30 30 30

2 1 2 3 6

30 30 30 30

3 2 4 6

4

30 30 30

12

30

1 2 3 6

30 30 30 30

P(y) 5

30

10 15

30 30

1

Then the marginal probability distribution at y = 3 is a)6 30 c) 12 30 b) 10 30

d) 15 30

105. If f(x,y) =  (6-x-y), 0 < x < 2, 2 < y < 4= 0, other wise Then marginal probability of x is a) (5-y) c) (3-x)  

b) (4-x) d) (4-y) 106. If f(x,y)= 2, 0 < x < y 0 =0, otherwise, determine f(x/y) a) e-y c) 0 d) 1 b)  (x+1) 108. Suppose F(x, y) be the joint distribution function for a continuous bivariate random variable x, y. Then, which of the following is incorrect? a) F (∞, ∞) = 1 b) F (∞, -∞) = 0 c) F(∞ , -∞) = 0 d) F (-∞, ∞) =1 109. Given the joint distribution function F(x,y) = 1 – e-x – e-y – e-(x+y), x > 0, y > 0 = 0, otherwise What is f(x)? c) e-x. e-y a) -e-x b) e-y d) ey 110. If F(x , x ) = k. .  , 0 0 111. If f(x,y) = e = 0, otherwise, then P (0 < x < 2, 1 0    b)  P[ k − µ ≤ kσ] ≥ 1 − 2 , k > 0   k k c) Both a and b d) None of these

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ANSWERS 1.d 2. c 3.b 4.b 5.c 6.a 7.b 8.d 9.c 10.c 11.a 12.b 13.a 14.a 15.c 16.d 17.c 18.b 19.d 20.c 21.c 22.b 23.b 24.d 25.d 26.d 27.b 28.d 29.a 30.a 31.c 32.b 33.a 34.c 35.c 36.b

37.d 38.d 39.c 40.a 41.b 42.b 43.b 44.d 45.d 46.c 47.b 48.c 49.a 50.a 51.c 52.b 53.c 54.d 55.c 56.b 57.b 58.d 59.d 60.c 61.b 62.c 63.a 64.c 65.b 66.c 67.a 68.b 69.a 70.d 71.c 72.b

73.d 74.b 75.b 76.c 77.c 78.c 79.c 80.c 81.c 82.b 83.b 84.d 85.b 86.c 87.d 88.a 89.b 90.d 91.a 92.d 93.a 94.a 95.c 96.c 97.b 98.b 99.c 100.c 101. b 102. c 103. b 104. d 105. c 106.b 107. b 108. d

109. a 110. c 111.d 112.a 113.c 114. b 115.a 116.c 117.a 118.c 119.a 120. c 121. c 122.a 123. b 124. b 125.d 126. a 127. a 128.d 129.b 130.a 131.c 132. a 133. b 134. c 135. a 136. b 137. d 138. c 139. a 140. d 141. c

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