THEORY OF EXPECTATION

THEORY OF EXPECTATION

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1) Expected value of a random variable 

A) is always positive 

B) may be positive or negative 

C) may be positive or negative or zero 

D) can never be zero 

2) If x and y are independent, then 

A) E(xy)= E(x) . E(y) 

B) E(xy)= E(x) + E(y)

C) E(x+y)= E(x) - E(y)

D) E(x - y)= E(x) + E(y)

3)  if all the values taken by a random variable are equal then 

A) its expected value is 0 

B) its standard deviation is zero 

C) its standard deviation is positive 

D) Its standard deviation is a real number .

4) if a and b are constants, then V(aX+b) equal to

A) aV(X) B) a²V(X) C) a²V(X)+C D)n 

5) If C is a constant then V(Cx) equal 

A) V(X) B) CV(X) C) C²V(X) D) n 

6) if variance of random variable X is 23, then what is the variance of 2x+10 ?

A) 56  B) 33    C) 46   D) 92 

7) If x and y random variable having expected  values as 4.5 and 2.5 respectively,  then the expected value of (x-y) is.. 

A) 2          B) 7        C) 6      D) 0 

8) Variance of random variable X is given by 

A) E(x- μ)²  B) E(x²) - {E(x)}²

C) E(x- μ²).   D) both A and B 

9) Four unbiased coins are tossed simultaneously. Let x be the random variable denoting the number of heads obtained. find the expectation of x, i.e., E(x)

A) 1   B) 2      C) 4   D) none 

10) taking data from the previous question,  what is the variance of the distribution. I e., V(x).

A)  1   B) 2.       C) 3         D) 6

*** A random variable x has the following distribution:

X:                   4       5    6        8
Probability: 0.1   0.3   0.4   0.2
Find 

11) the expectation of x 

A) 4.58 B) 5.01 C) 5.9 D) 6.4 

12) standard deviation of x.

A) 5.99 B) 1.01 C) 2.25  D) 1.22 

** X:               4       6        7      10
Probability: 0.2    0.4    0.3     0.1
Find: 

13) expectation of x 

A) 5.9.  B) 6.3. C) 1.2 D) 6.9 

14) Standard deviation of x 

A) 1.22. B) 1.11. C) 1.62  D) 2.61 

15)  X:.        -1      0      1      2
Probability: ⅓      ⅙     ⅙    ⅓
Find the expectation of X. 

A) 0.33. B) 0.45  C) 0.25. D) 0.50 

** X:    0   1    2     3    4   5     6     7
P(x):    0  2k  3k    k  2k   k² 7k² 2k²+k
16) What the value of k ?

17) value of P(x<6)
18) value of P(x>6)
19) value of P(0<x<5)

20) In a business venture, a man can make a profit of Rs2000 with a probability of 0.4 or have a loss of Rs1000 with a probability 0.6. what is his expected profit ?

21) A player tosses 3 coins. He wins Rs16. If 3 heads appear, Rs8 if two heads appear, Rs4 if one head appears and Rs2 if no heads appear. Find his expected amount of winning.

22) Three coins whose faces are marked 1,2 are tossed. What is the expectation of the total value of Number on their faces.

A) 36 B) 38 C) 4.5 D) 5.5

***The monthly demand for radio is known to have the following probability distribution:
Demand:      1      2       3     4       5
Probability: 0.1  0.15  0.2  0.25 0.2
23) What is the expected demand for radio ?
24) what is the variance of the distribution of demand for radio?
25) If the cost of producing radios is given by C= 1000+2n, determine the expectation cost.

26) A Number is chosen at random from the set 1,2,3….100 and another Number is chosen at random from the set 1,2,3...50. what is the expected value of their product.

27) A person tosses a coin once and is to receive Rs4 for head and is to loss Rs2 for tail. Find the expectation and variance of his gain ?

28) A person tosses two coins simultaneously and is to receive Rs8 for two head, Rs2 for one head and he is to pay Rs6 for no head . Find his expectation.

29) A man draws 2 balls from a bag containing 3 white and 6 black balls. If he is to receive Rs14 for every white ball and Rs7 for every black balls. What is his expectation.

30) The probability that there is atleast one error in an accounts statement prepared by A is 0.2 for B and C it is 0.25 and 0.4 respectively. A,B,C prepared 10,16,20 statement respectively. Find the expected number of correct statement in all.

31) A box contains 8 tickets, 3 of the tickets carry a prize of Rs5 each and other 5 carry a prize of Rs2 each.what is the expected value when one ticket is drawn at random from the box ?

32) An unbiased coins is tossed four times, if X denotes the number of heads, Calculate the expected value and variance of x.

33) If rains a taxi driver can earn Rs100 per day if it is fair, he can lose Rs10 per day. If the probability of rain is 0.4 what is his expectation?

34) Throwing 2 unbiased coins simultaneously, Mr. X bets with
Mrs. X that he will receive Rs4 from her if he gets 2 head and he will give Rs4 to her otherwise. Find Mr. X's expectations ?

35) A player tosses 3 fair coins. He wins Rs10 if 3 head appear. Rs6 if 2 head appears and Rs2 if 1 head appears. On the other hand he loses Rs25 if 3 tails appears. Find the expected gain of the player.

36) A card is drawn at random from a pack of 52 cards. If ace counts one, king, queen and Jack count 10 each and other count at their face value; find the EXPECTATION of the value of the cards.

37) A bag contains 5 white and 7 black balls. Find the expectation of a man, who is allowed to draw two balls from the bag and who is to receive one rupee for each black ball and two rupees from each white ball drawn.

38) A and B play for a prize of Rs99. The prize is to be won by a player who first throw a "3" with one dice. A first throws and if he fails, B throws and if he fails again throws and so on. Find their respective expectation.(A's expectation and B's expectation)

39) If three coins are tossed. The expectation of the number of heads is:

40) A die is thrown at random. The expectation of the number on it is:

41) In a random throw of 2 dice is the expectation of the sum of the points on them is :

42) In a random throw of 2 dice, the expectation of the product of points on them is:

43) A throws a coin 3 times. If he a head all the three times he is to get a prize of Rs160. The entry fee for the game is Rs16. The Mathematical expectation of A is -

44) If rains a dealer in umbrella can earn Rs300 per day, if it does not rain he can lose Rs80 per day. What is his expectation if the probability of a rainy day is 0.57(in rupee) is:

45) A box contains 8 items of which 2 are defective. A man selects 3 items at random. The expected number of defective items he has drawn is -

46) A player tossed two coins. If two heads show he wins Rs4. If one head shows he wins Rs2, but if two tails show he pays Rs3 as penalty. The expected value of the game to him (in rupee) is:

47) The expected value of X, the sum of the scores when two dice are rolled is :

48) Three items are drawn at random from a box containing 2 defective and 6 non-defective items. The expected number of non-defective items drawn is:

49) A player tosses 3 fair coins. He wins Rs5 if 4 heads appear, Rs3 if 2 heads appear, Rs1 if 1 head occurs. On the other hand, he loses Rs15 if 3 tails occur. His expected gain is:

50) A player tosses three coins. He wins Rs8 if 3 heads appear, Rs3 if 2 heads appear, and Rs1 if only 1 head appear. If the game is fair, how much would he lose if no head occur.

51) Two cards are selected at random from a box, which contains five cards numbered 1,1,2,2 and 3. Let X denote the sum of Numbers. The expected value of the sum is:

52) An urn contains 7 white and 3 red balls. Two balls are drawn together at random from this urn. The expected number of white balls drawn is:

53) There are six slips in a box and Numbers 1,1,2,2,3,3 are written on these slips. Two slips are taken at random from the box. The expected values of the sum of Numbers on the two slips is.