Question 1

Let X be random variable with probability density function

Which of the following statement is correct?

Accepted Answer

mean exists but variance does not exist

Answer

both mean and variance do not exist

Answer

variance exists but Mean does not exist.

Answer

both mean and variance exist

Question 2

A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is

The mean and variance of the shorter of the two pieces are respectively

Accepted Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Question 3

Consider a game where the player tosses a six-sided fair die.

If the face that comes up is 6, the player wins 36 rupees , otherwise he loses $k^2$ rupees ,

where k is the face that comes up k ={1,2,3,4,5}.

The expected amount to win at this game in rupees is

Accepted Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Question 4

A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

Accepted Answer

1

Answer

3

Answer

2

Answer

4

Question 5

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

Accepted Answer

2

Answer

3

Answer

6

Answer

4

Question 6

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are

Accepted Answer

2i–n, i = 0,1,2...n

Answer

n–i, i = 0,1,2...n

Answer

2i+2n, i = 0,1,2...n

Answer

i+2n, i = 0,1,2...n

Question 7

If the function $f\left(x\right)\ for\ a represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?$

Accepted Answer

16 and 24

Answer

7 and 19

Answer

0 and 12

Answer

5 and 17

Question 8

Four buses carrying 160 students from the same school arrive at a football stadium. The busescarry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student.One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus.

Then E(X) and E(Y) respectively are

Accepted Answer

50,40

Answer

40 ,50

Answer

41,41

Answer

40.75, 40

Question 9

Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with Probability 0.5. Assume that the results of the flips are independent, and let X equal the total number of heads that result. The value of E(X) is

Accepted Answer

0 11

Answer

1.1

Answer

11

Answer

1

Question 10

On a multiple-choice exam with 3 possible destructives for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is

Accepted Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Question 11

If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0) is

Accepted Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Question 12

If X is a binomial random variable with expected value 6 and variance 2.4, then P(X = 5) is

Accepted Answer

$\frac{l}{2},\frac{l^2}{12}$ $\left(\frac{3}{5}
ight)^6\left(\frac{2}{5}
ight)^4$

Answer

$\frac{l}{2},\frac{l^2}{12}$ $\left(\frac{3}{5}
ight)^6\left(\frac{2}{5}
ight)^4$

Answer

$\frac{l}{2},\frac{l^2}{12}$ $\left(\frac{3}{5}
ight)^6\left(\frac{2}{5}
ight)^4$

Answer

$\frac{l}{2},\frac{l^2}{12}$ $\left(\frac{3}{5}
ight)^6\left(\frac{2}{5}
ight)^4$

Question 13

The random variable X has the probability density function

and E X( ) = 712, then a and b are respectively

Accepted Answer

1 and $f\left(x\right)\ for\ a represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?$

Answer

1 and 2

Answer

$\frac{l}{2},\frac{l^2}{12}$ and 1

Answer

2 and 1

Question 14

Suppose that X takes on one of the values 0, 1, and 2.

If for some constant k, P( X=i) = kP(X=i-1) for i=1,2 and P (X=0) = $k^2$ , then the value of k is

Accepted Answer

1

Answer

3

Answer

4

Answer

2

Question 15

Which of the following is a discrete random variable?

I. The number of cars crossing a particular signal in a day.

II. The number of customers in a queue to buy train tickets at a moment.

III. The time taken to complete a telephone call.

Accepted Answer

I and II

Answer

III only

Answer

II only

Answer

II and III

Question 16

If the function is a probability density function of a random variable, then thevalue of a is

Accepted Answer

1

Answer

2

Answer

4

Answer

3

Question 17

The probability mass function of a random variable is defined as:

Then E(X ) is equal to:

Accepted Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Question 18

Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X–3) is

Accepted Answer

0.24

Answer

0.6

Answer

0.48

Answer

0.96

Question 19

If in 6 trials, X is a binomial variable which follows the relation 9P(X=4) = P(X=2), then the probability of success is

Accepted Answer

0.125

Answer

0.75

Answer

0.25

Answer

0.375

Question 20

A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom.

What is the probability that he will sell a computer to exactly two of the next three customers?

Accepted Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

Answer

$\frac{l}{2},\frac{l^2}{12}$

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