probability theory & stochastic processes _bits

7/30/2019 Probability Theory & Stochastic Processes _BITS

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Balaji Institute of Engineering & Science

Probability Theory & Stochastic Processes

Quiz -1 Bits

Branch: II ECE--------------------------------------------------------------------------------------

1. If the events A and B are mutually exclusive, then P(A or B) isa) P(A) + P(B) b) P(A) - P(B) c) P(A) P(B) d) P(A) + P(B) P(AB)

2. The probability of drawing a red ace from a pack of 52 cards isa) 4/52 b) 2/52 c) 1/52 d) 1

3. Two events X and Y are statistically independent then, P(XY) isa) P(X) + P(Y) b) P(X) P(Y) c) 0 d) 1

4. Three cards are drawn from an ordinary deck of 52 playing cards. The probability that all threecards are even numbered of black color is --------------------.

a) 0.054 b) 0.0054 c) 0.54 d) 0.0000545. Each value in the sample space of a phenomenon is called as ---------------.6. Let x be the outcome from rolling one die and y the outcome from rolling a second die. The joint

probability of the event X3 and Y>3 is ------------------------.

7. The probability of getting three 3s and then 4 or 5 in four rolls of a balanced die is -------------.8. Two fair dice are thrown simultaneously. Let X and Y are the numbers on the first and second die.

Then P(X=Y) = ----------------.a) 6/36 b) 8/36 c) 10/36 d) 1/36

9. The probability of drawing a red ball from a bag containing 6 red balls and 3 black balls is -------.a) 7/9 b) 6/9 c) 10/12 d) 1

10. The probability of drawing an ace or a card of clubs from a pack of 52 cards is -------- .a) 17/52 b) 16/52 c) 18/52 d) 20/52

11. Two coins are tossed together. The probability of MATCH will be ---------------.a) 1/4 b) 1/4 c) 0 d) 1

12. If the events A and B are independent, then P(A/B) isa) P(A) b) P(A/B) c) P(A) P(B) d) P(B)

13. Two dice are thrown. The probability of getting sum equal to 12 isa) 5/36 b) 5/52 c) 1/36 d) 1

14. If two dice are thrown, what is the probability of not throwing a total of 4?a) 33/36 b) 4/36 c) 11/36 d) 12/36

15. The value of the probability of an event lies in between ------- and ----------.a) 0,1 b) -1,0 c) -, + d) -1, +1

16. Probability of an sure/certain event isa) b) 0 c) - d) 1

17. Probability of an impossible event isa) b) 0 c) - d) 1


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18. Always, the probability of an event is a non negative quantity. [ True/False ]19. If the events A and B are mutually exclusive, then P(AB) is

a) P(A) + P(B) b) 0 c) P(A) P(B) d) P(A) + P(B) P(AB)

20. A husband and his wife appear in an interview for two vacancies in the same post. The probabilityof husbands selection is 1/7 and that of wifes selection is 1/5. the probability that both of them

will be selected isa) 3/35 b) 2/35 c) 1/35 d) 1

21. ------ is an experiment whose outcome or result is not unique and therefore cannot be producedwith certainty.

a) random experiment b) deterministic experiment

c) non deterministic experiment d) both a and c

22. If two coins are tossed simultaneously, the probability of getting two heads will bea) 2/4 b) 1/4 c) 3/4 d) 1

23. Each letter of the word ATTRACT is written on a separate card. The cards are then thoroughlyshuffled, and four of them are drawn in succession. The probability of getting result TACT is ----.

24. A card is drawn from a deck of 52 cards. What is the probability that 2 of clubs is drawn?25. A card is drawn from a pack of cards. Find the probability of it being not a spade.26. A set of events is said to be ------ if one of the events cannot be expected in preference to any

other.a) equally likely b) mutually exclusive c) mutually exhaustive d) none

27. A set of events is said to be ---- if the occurrence of one of the events excludes the possibility ofthe occurrence of any other.

a) equally likely b) mutually exclusive c) mutually exhaustive d) none

28. Two dice are thrown, the probability of getting a sum of 5 isa) 2/9 b) 3/9 c) 1/9 d) 1

29. A set of events is said to be ----- if the occurrence of an event is not influenced by the occurrenceof any of the rest of the set.

a) dependent b) independent c) equally likely d) mutually exclusive

30. If A be the event and _A be the complementary event then P(A)+P( _A ) =a) P(A) b) P(

_

A ) c) 1 d) 0

31. A problem is given to three students A, B, C whose probabilities of solving it are , 1/3 and respectively. What is the probability that the problem will be solved.

a) 1/4 b) 3/4 c) 2/4 d) none

32. A die is tossed. If the number is odd, what is the probability that it is prime?a) 1 b) 2/3 c) 1/3 d) none

33. If the probability distribution function of a continuous Random Variable is a ramp function, whatwill be its density function?34. If the probability distributed function of a Random Variable X is a step function with magnitude of

1/2 , then its probability density function isa) step function with magnitude of 1/2 b) impulse function with magnitude of c) step function with magnitude of 1/4 d) impulse function with magnitude of


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35. The minimum value that CDF of a Random Variable can have is ---- and the maximum value is ---.a) 0, 1 b) -, 0 c) 0, d) -,

36. The value of constant K is ----------------------, so that the following is probability den sity function.fx = K (1-x

2) for 04) isa) 3/4 b) 3/7 c) 3/5 d) none

41. FXY(x,) = ------------.a) FX(x) b) 0 c) d) -1

42. The area under Gaussian pdf curve isa) zero b) 1 c) d) -1

43. The Gaussian pdf -------- shaped curve.a) rectangular b) triangular c) bell d) square

44. pdf, in terms of CDF is given asa) derivative of CDF b) integral of CDF c) logarithmic of CDF d) all of the above45. Let X and Y are two continuous Random Variables, then the joint probability function of X and Y is

defined by -------------------.

46. Random Variable is also referred to asa) stochastic variable b) variate c) both a and b d) none

47. The basic classification of Random Variables is as follows:1. Discrete Random Variable and continuous Random Variable2. Dependent Random Variable and independent Random Variable.Of these statements,a) only 1 is correct b) only 2 correct c) both 1 and 2 correct d) none

48. If a Random Variable takes an infinite number of uncountable values, it is called ---- RandomVariable.

a) continuous b) discrete c) both a and b d) none

49. If a Random Variable takes a finite set of values, it is called ------------ Random Variable.a) dependent b) independent c) continuous d) discrete


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50. For a discrete Random Variable, plot of CDF is aa) impulse form b) sinusoidal form c) staircase form d) none

51. If the probability distribution function of a Random Variable X is unit step function, its pdf isa) unit impulse function b) power function c) energy function d) none

52. Cumulative distribution function of Random Variable X is defined as FX(x) = ---------.a) P(X=x) b) P(Xx) c) P(Xx) d) none

53. The probability distribution of X is given byX=xi 1 2 3 4

P(xi) 1/4 1/4 1/4

Then FX(2)=----------------.a) 1/4 b) 1/2 c) d) 1

54. FX(-) = -------------------.a) 0 b) 1 c) d) none

55. FX() = -------------------.a) 0 b) 1 c) d) none56. P(x1


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67. For a standardized normal Random Variable, mean and variance are respectively------------.a) 0 and 1 b) 1 and 0 c) 0 and 0 d) 1 and 1

68. If P(X=xi)=1/P for i=1, 2, 3,.P, then the Random Variable X is said to have----- probabilitydistribution.

a) Uniform b) Non uniform c) Normal d) none

69. Complementary error function is also referred to as co-error function. This statement isa) true b) false c) data insufficient d) none70. The joint distribution function FX,Y(x,y) is a --------------- function of X and Y.

a) decreasing b) non decreasing c) increasing d) none

71. A Random Variable X has the following probability distributionX=xi 0 1 2 3 4

P(xi) 0 K 2K 2K 3K

Then the value of K isa) 1/7 b) 1/8 c) 1/6 d) none

72. The probability function of a Random Variable X is given byf(x) = 2K for X=1= 3K for X=2= 3K for X=3= 0 otherwise where K is a constant, then P(1X2) is

a) 6/8 b) 5/8 c) 1 d) none

73. The joint probability function of two discrete Random Variables X and Y is given byf(x,y) = C(2x+y) for 0x2 and 0y3

= 0 otherwiseThen X and Y are

a) independent b) not independent c) none

74. If pdf of X is f(x) and pdf of Y is f(y), then the pdf of X+Y isa) f(X) + f(Y) b) f(X)/f(Y) c) f(X).f(Y) d) f(X)*f(Y)

75. Mathematical of classical or priori probability definition fails whena) the outcome are not equally likely b) number of outcomes is infinitec) both a and b d) none

76. Which of the following is the parameter of the normal distribution.a) mean b) standard deviation c) both a and b d) none

77. The joint density function of two continuous Random Variables isf(x, y) = Cxy; 0


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79. Which of the following is the example of probability density function?a) Gaussian probability density b) Rayleigh probability densityc) both a and b d) none

80. The joint density function of two continuous Random Variables X and Y is given by

yxeyxf

yx,;

4

1),(

Then X and Y are ------- Random Variables.a) statistically dependent b) statistically independent c) none

81. The value of b so that the function is a valid density function.

otherwise

bxexfx

;0

0;4

1)(

3

a) 0.58 b) 0.855 c) 1 d) none

82. A Random Variable Uniformly distributed over (-a, a) has a pdf given bya) 1/4a b) 1/3a c) 1/2a d) 1/a

83. If two Gaussian Random Variables are added then the resulting variable isa) Gaussian b) Rayleigh c) rectangular d) none

84. Many of the naturally occurring experiments are characterized by ------ distribution.a) Poisson b) Binomial c) Gaussian d) none

85. The Rayleigh distribution is based on two independenta) Poisson variables b) exponential variables c) Gaussian variables d) none

86. The maximum value of Gaussian distribution isa)

2

1b)

E

1c)

2

1d) none

87.

The expectation of a Random Variable is equal to itsa) variance b) standard deviation c) mean d) none

88. K be a constant then E[K] =a) K b) 0 c) 1 d) none

89. K be a constant then Variance(K) isa) 0 b) 1 c) 2 d) none

90. The pdf of Gaussian distribution with mean=0, variance=1a) 2

2

1x

e

b) 2

2

2

1x

e

c) 2

2

2

1x

e

d) none

91. Let X be a uniformly distributed Random Variable defined 4x8, then its pdf isa) 1/2 b) 1/(b-a) c) d) none

92. X and Y are two Random Variables, then V(X+Y) isa) V(X)+V(Y) b) V(X)-V(Y) c) 1 d) none

93. If X is a Random Variable and a be a constant then V(aX) isa) V(X) b) a2V(X) c) V(X) d) none


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94. If X and Y are two independent Random Variables and 2x =4, 2y =9, then V(4X+2Y) isa) 100 b) 0 c) 120 d) none

95. Expectation of the number on a die, when a die is throwna) 2/7 b) 7 c) 1 d) 7/2

96. If two dice are rolled, then what is the expectation of the product of the numbers on the dice.a) 49/4 b) 7/2 c) 7 d) none

97. Moments generating function of a Random Variable is used to generate moments about ----- of theRandom Variable.

a) origin b) mean c) variance d) none98. The second moment of a Random Variable about its mean gives ---- of the Random Variable.

a) 0 b) 1 c) variance d) none

99. The second moment of Random Variable about origin givesa) mean b) variance c) mean square value d) none

100. The variance is alwaysa) positive b) negative c) both a and b d) none

101. If X is Gaussian Random Variable with mean zero then pdf isa)

2)(

2

1

mx

e

b))

2(

2

2

2

1

x

e

c))

2(

2

2

1x

e

d) none

102. If X and Y are orthogonal thena) E[XY]=0 b) V(X)=0 c) V(Y)=0 d) none

103. If X and Y are independent then conditional densities are equal toa) marginal density b) joint density c) 0 d) none

104. If X and Y are independent then E[y/X=x] isa) E[Y] b) E[X] c) E[XY] d) none

105. The Gaussian distribution is symmetric about itsa) variance b) mean c) standard deviation d) none

106. If the pdf of a Random Variable X is f(x)=e-x, x0=0 otherwise.

Then E[X] isa) 0 b) -1 c) 1 d) none

107. If X and Y are independent Random Variables and E[X]=7/12, E[Y]=7/12 then E[XY] isa) 0 b) 1 c) -1 d) 49/144

108. If X and Y are two independent Random Variables and V(X)=4, V(Y)=9 then V(2X+3Y) isa) 13 b) 97 c) 35 d) 0

109. If X is a Random Variable, a being a constant then V(X+a) isa) V(X) b) V(X) + V(a) c) V(X) - V(a) d) none

110. If X is Random Variable, a being a constant then E[X+a] isa) E[X] b) E(X)+E[a] c) E(X)+a d) none

111. If X and Y are two independent Random Variables and E[X]=2, E[Y]=3, V(X)=1 and V(Y)=4 thenE[X+Y] and V(X+Y) is

a) 2,1 b) 3,4 c) 1,1 d) 5,5


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112. If f(x) = e-x for x0= 0 for x2, then E(x/3) isa) 0 b) 2 c) 4/3 d) none

114. Let X be a continuous Random Variable with pdf f(x), the transformation y=g(x) is said tomonotonic decreasing if

a) g(x1) < g(x2) for x1 g(x2) for x1g(x2) for x1>x2 d) none

115. If X be a continuous Random Variable with pdf f(x), and y=g(x) then f(y) isa) f(x) b) f(y) (dy/dx) c) f(x)|dx/dy| d) none

116. The standard deviation of a Random Variable X isa) )(xV b) )(xV c) )(xV d) none

117. The maximum magnitude of characteristic function isa) 0 b) 1 c) -1 d) none

118. If f(x) is a pdf of Gaussian Random Variable, then the maximum value of f(x) occurs at x=a) mean b) 0 c) 1 d) 2119. The moments determined about the mean value are called as -----------.120. If X be the outcome from rolling one die and Y be the outcome from rolling a second die, then

E[X/Y] is -----------------------.

121. If the two Random Variables are independent i.e., E[XY]=E[X] E[Y]; then, the Random Variablesare --------------------------.

122. If K is any constant, then E[KX) isa) E[X] b) K E[X] c) K2 E[X] d) none

123. If X and Y are any two Random Variables, then E[2X+3Y] isa) 2 E[x] + 3 E[Y] b) 6[E[x] + 3 E[Y]] c) 6E[XY] d) none

124. If X and Y are independent Random Variables, then E[6XY] isa) 6 E[X] b) 6 E[Y] c) 6 E[X] E[Y] d) none

125. If X is a Random Variable, then E[3X+6] isa) 3 E[X] b) 6 E[X] c) 3 E[X] + 6 d) none

126. If pdf f(x) of a Random Variable is an even function thatis f(-x) = f(x) then E[X] isa) 0 b) 1 c) 2 d) none

127. If the pdf f(x) of a Random Variable X is symmetrical about the point b that is f(b-x)=f(b+x), thenE[X] is

a) 1 b) 0 c) 2 d) b

128. The variance is also called as ------------ of the Random Variable X.a) first central moment b) second central moment c) third central moment d) none

129. Mean or the expectation of a Random Variable X is also called as ----- about origin.a) first moment b) second moment c) third moment d) none

130. The variance or dispersion of a Random Variable X is given by 2 =-----.a) E[X2] (E[X])2 b) E[X2] E[X] c) E[X]2 -E[X2] d) none


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131. Which of the following is used as measure of the spread of a Random Variable about its mean?a) moment generating function b) characteristic functionc) standard deviation d) none

132. If X is a Random Variable then Var(-2X) isa) -2 Var(X) b) 2 Var(X) c) 4 Var(X) d) none

133. In the expansion of moments generating function the coefficient of t2/2! Is equal toa) mean b) variance c) mean square value d) none134. Characteristic function of a Random Variable is X(t), if t=0 then X(t) is

a) 0 b) -1 c) 1 d) none

135. If a Random Variable X is such thatE [ (X+1)2 ] = 4, E [ (X-1)2 ] = 2. then E[X] is

a) b) 2 c) d) 4

136. Consider the following statements.1. Characteristic function always exists even though the moment generating function doesnotexist.2. Moment generating function always exists even though the characteristic function doesnot exist.

Of these statementsa) only 2 is correct b) only 1 is correct c) both are correct d) none

137. Two independent Random Variables are ---- Random Variables.a) correlated b) uncorrelated c) none

138. Which one of the following statement is correct.a) density of sum of two Random Variables is equal to the convolution of individual densities of theRandom Variables.b) density of sum of two Random Variables is equal to the sum of individual densities of theRandom Variablesc) density of sum of two Random Variables is equal to the product of individual densities of theRandom Variables.d) none

139. A Random Variable X has the density function.f(x) = x2 for -1


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145. The rth moment of a continuous Random Variable X about the mean m isa)

dxxfmx r )()( b)

dxxfmx )()(

c)

dxxfmx r )()( d)

dxmx r)(

146. The Random Variable X follows Poisson distribution such that 2P(X=0)=P(X=1), then the mean ofdistribution is

a) 4 b) 2 c) 2 d) 21/4

147. The Random Variable X has a Poisson distribution. If P(X=1)=2 and P(X=2)=4 then the standarddeviation of X is

a) e-1 b) e-2 c) e-4 d) e-16

148. A Random Variable X with no moment generating function existing will have its characteristicfunction-------.

a) non existing b) existing c) none

149. Two Random Variables X and Y with identical moment generating functions are ---- distributed.a) identically b) differently c) none

150. Let X(t) is a characteristic function of Random Variable X, if Y=aX+b then Y(t) isa) eibt.X(at) b) e

ibt.X(bt) c) X(t) d) none

151. IfX(t) is a characteristic function then | X(t) | isa) 1 b) c) equal to 1 d) none

152. If X is a Random Variable with moment generating function MX(t), then the moment generatingfunction of Y=3X+2 is

a) e3t MX(2t) b) e2t MX(3t) c) e

3t MX(t) d) e2t MX(t)

153. If X and Y are two independent Random Variables with moment generating functions MX(t) andMY(t). then MX+Y(t) is

a) MX(t) + MY(t) b) MX(t) . MY(t) c) 0 d) none

154. If X is a Random Variable and its MGF is MX(t)=1/(1-t), then 1st moment (m1) isa) 1 b) 2 c) -1 d) none

155. A fair die is tossed 30 times. A toss is called a success if face 1 or 2 appears. Then the mean forthe number of successes is

a) 30 b) 20 c) 90 d) 10

156. Which of the following theorem states that the probability density of a sum of N independentRandom Variables tends to approach a normal density as the number N increases

a) Bayes theorem b) Central limit theorem c) Channel capacity theorem d) none

157. Pick the odd man outa) stochastic variable b) stochastic function c) Random Variable d) random experiment

158. Pick the odd man outa) binomial distribution b) normal distribution c) uniform distribution d) Rayleigh distribution

159. Which of the following is incorrect?a) P(S)=1 b) P(A)=P(A)-1 c) 0P(A)1d) if A and B are mutually exclusive, then P(A+B)=P(A)+P(B)


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160. Pick the odd man outa) expectation b) variance c) standard deviation d) Chebycheffs inequality

161. Which of the following is correct?a) unit of variance is same as that of the Random Variableb) the probability that a continuous Random Variable takes on a particular value is zeroc) mutually exclusive events are also statistically independent

d) the classical approach for probability theory does not explain the situation when the number ofoutcomes of an experiment is small.

162. The first moment of a Random Variable X about its mean isa) zero b) mean c) mean square value d) variance

163. For mX=0, the variance is same as -------- of the Random Variable.a) zero b) mean c) mean square value d) variance

164. The joint CDF of two independent Random Variables X and Y can be expressed as fXY(x, y)=a) fX(x).fY(y) b) fX(x)+fY(y) c) fX(x)-fY(y) d) none

165. Characteristic function is used to find moments about ---- of a Random Variable.a) mean b) origin c) none

166. IfX() is the characteristic function of the Random Variable X, then X()|=0isa) 0 b) 1 c) d) none

167. Moment generating function of a Random Variable is used to generate moments about ---- of theRandom Variable.

a) origin b) mean c) both a and b d) none

168. If X and Y are independent then moments generating function of (X+Y) isa) MX(t).MY(t) b) MX(t)+MY(t) c) MX(t)/MY(t) d) none

169. If X and Y are independent then the characteristic function of (X+Y) isa) X(t)+Y(t) b) X(t)/Y(t) c) X(t).Y(t) d) none

170. The MGF of X is MX(t) and MGF of Y is MY(t), if X and Y are identical thena) MX(t)=MY(t) b) MX(t)MY(t) c) MX(t).MY(t) d) none

171. Characteristic function is defined as X()= -----a) E[ejx] b) E[e-jx] c) E[ejx] d) none

172. Moment generating function of a standardized normal variable isa)

2/te b)2t

e c)2/

2t

e d)t

e

173. If X and Y are two independent variables i.e., E[XY] = E[X].E[Y] and hence XY isa) 0 b) c) 1 d) none

174. If X and Y two Random Variables and X=Y then the correlation coefficient isa) 0 b) c) 1 d) none

175. If X and Y are two independent Gaussian variables, with means mx=2, my=2,2

x =4,

2

y =9 thenmean and variance of X+Y is

a) 2, 2 b) 0, 1 c) 5, 13 d) none

176. if X and Y are independent the COV(x, y) isa) 1 b) 2 c) 1/2 d) 0


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177. if X and Y are independent then correlation coefficient isa) -1 b) 1 c) 1/2 d) 0

178. Correlation coefficient is always lies betweena) 0 and 1 b) 0 and 2 c) -1 to 1 d) none

179. If X and Y are two independent Random Variables, and a and b are constants then V(aX+bY) isa) a2V(X) + b2V(Y) + 2ab CoV(XY) b) a2V(X) + b2V(Y) - 2ab CoV(XY)

c) V(X) + V(Y) + 2ab CoV(XY) d) a2V(X) + b2V(Y)

180. The correlation coefficient of two Random Variables X and Y is while their stan dard deviationsare 2 and 4 then the covariance of X and Y is

a) 2 b) 8 c) 4 d) none

181. Cov(4X, 3Y) isa) 4 Cov(X,Y b) 7 Cov(X, Y) c) 12 Cov(X, Y) d) none

182. The correlation coefficient of two Random Variables X and Y is -1/4 while their variances are 3 and5 the XY is

a) 1 b) 0 c)4

15d) -

4

15

183.

If X and Y are two Random Variables, a and b are constants then CoV(aX, bY) isa) CoV(X, Y) b) a2b2CoV(XY) c) (a/b) CoV(X, Y) d) ab CoV(X,Y)

0 1 2 3 4 5 6 7 8 9

0 a b b b a b

1 b b a c a d b T b

2 c d b a b c b

3 c b b b a T F

4 a b c a c a a D

5 c a b b a b b b b A6 a c a a a a b a a A

7 b b b a d c c a b C

8 b b c a c c a c a A

9 b c a b a d a a c C

10 a b a a a b c d b A

11 c d d b c b b a

12 b a c c a d b A

13 a c c a c b b a c A

14 b d c a b a c c b A

15 a a b b a d b d a B

16 d b a c a b b a a C17 a a c a c c d d c D

18 c c d d

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