# Random Variables and Distribution Functions MCQ Questions

Please go through Random Variables and Distribution Functions MCQ with Answers for Environment Economics provided below. Students should have strong knowledge about the Random Variables and Distribution Functions as in various competition exams, MCQ questions are asked from this topic. We have provided below the biggest collection of Random Variables and Distribution Functions MCQ with Answers. These Business Statistics MCQ and objective questions will improve your performance in exams and help you to get good scores.

## Multiple Choice Questions for Random Variables and Distribution Functions with Answers

Question. For what values of k can f(x) = (1-k) kx
a) 0<k<1
b) k=0
c) k>1
d) None of these

A

Question. In a business venture a man can make a profit of Rs 2,000 with probability of 0.4 or have a loss of Rs 1,000 with a probability of 0.6. His expected profit is
a) 100
b) 200
c) 400
d) 300

B

Question. From a bag containing 4 white and 6 red balls, three balls are drawn at random and if each white ball drawn carries a reward of Rs4 and each red ball Rs6, find the expected reward of the draw
a) Rs14.8
b) Rs15.6
c) Rs31
d) Rs16

B

Question. If X1 and X2 are independent random variables having exponential densities with the parameters a and b the probability density of Y = X1+ X2 when a ≠ b
a) f(y) = 1/a+b. (e-y/a – e-y/b ) for y > 0 and f(y) = 0 elsewhere
b) f(y) = 1/a-b. (e-y/a – e-y/b ) for y < 0 and f(y) = 1 elsewhere
c) f(y) = 1/a-b. (e-y/a – e-y/b ) for y > 0 and f(y) = 0 elsewhere
d) None of these

C

Question. If X has a hypergeometric distribution with M = 3, N = 6 and n = 2, find the probability distribution of Y, the number of successes minus the number of failures
a) h(0) = 1/5 , h(1) = 3/5 , h(2) = 1/5
b) h(0) = 2/5 , h(1) = 3/8 , h(2) = 1/5
c) h(0) = 9/5 , h(1) = 3/5 , h(2) = 1/5
d) h(0) = 1/5 , h(1) = 4/5 , h(2) = 1/5

A

Question. Obtain the probability distribution of the number of sixes in 2 tosses of dice
a) x/p(x), 0/(4/9), 1/(4/9), 2/(1/9)
b) x/p(x), 0/(4/72), 1/(1/9), 2/(1/9)
c) x/p(x), 0/(4/9), 1/(4/36), 2/(8/9)
d) x/p(x), 0/(25/36), 1/(10/36), 2/(1/36)

D

Question. Three cards are drawn at random successively, with replacement, from a well shuffled pack of cards. Getting a card of ‘diamonds’ is termed as success. Obtain the probability distribution of the number of successes.
a) x/p(x), 0/(27/64), 1/(27/64), 2/(9/64), 3/(1/64)
b) x/p(x), 0/(1/9), 1/(4/9), 2/(1/9), 3/(6/9)
c) x/p(x), 0/(4/9), 1/(4/9), 2/(1/9), 3/(5/9)
d) x/p(x), 0/(4/64), 1/(7/64), 2/(1/64), 3/(8/64)

A

Question. A number is chosen at random from the set 10.11,12- – -109; and another number is chosen at random from the set 12,13 ,14- – – 61. The expected value of their sum is
a) 95
b) 96
c) 97
d) 98

B

Question. Three coins whose faces are marked 1 and 2 are tossed. Their expectations of the total values of numbers on their faces is
a) 9.5
b) 4.5
c) 3
d) 4

B

Question. A die is thrown at random. What is the expectation of the number on it:
a) 3.7
b) 3.1
c) 3.5
d) 3.8

C

Question. What is the expected number of heads appearing when a fair coin is tossed three times?
a) 2.1
b) 1.5
c) 3.2
d) 4.1

B

Question. If the probability density of X is given by

and Y = X2
The probability density of Y is
a) g(y) = e-y for y > 0 and g(y) elsewhere
b) g(y) = ey for y > 0 and g(y) = 0
c) g(y) = e-y for y< 0 and g(y) > 0
d) None of these

A

Question. If X has the uniform density with the parameters α = 0 and β = 1. Find the probability density of the random variable Y = √X
a) g(y) = y for 0 < y < 1 and g(y) = 0 elsewhere
b) g(y) = 2y for 0 < y < 1 and g(y) = 0 elsewhere
c) g(y) = 2y for 0 > y > 1 and g(y) = 0 elsewhere
d) None of these

B

Question. A contractor spends Rs. 3,000 to prepare for a bid on a construction project which, after deducting manufacturing expenses and the cost of bidding, will yield a profit of Rs. 25,000 if the bid is not won. If the chance of winning the bid is 10%, compute his expected profit?
a) 100
b) 607
c) 35
d) 200

D

Question. Determine which of the following given values can serve as the values of a probability distribution of a random variable with the range x = 1, 2, 3 and 4
a) f(1) = 0.25 , f(2) = 0.75 , f(3) = 0.25 , f(4) = -0.25
b) f(1) = 0.15 , f(2) = 0.27 , f(3) = 0.29 , f(4) = 0.29
c) f(1) = 1/19 , f(2) = 10/19 , f(3) = 2/19 , f(4) = 5/19
d) None of these

B

Question. A lot of 12 television sets include 2 with white chords. If 3 of the sets are chosen at random for shipment to the hotel, how many sets with white chords can the shipper expect to send to the hotel
a) 0
b) 1
c) 1/2
d) All of the above

C

Question. If the probability density of X is given by

To evaluate E[(2X+1)2]
a) 2
b) 1
c) 4
d) 3

D

Question. A and B throw with one die for a prize of Rs199 which is to be won by the player who first throws 6. If A has the first throw their respective expectation are
a) Rs 64, Rs 46
b) Rs 54, Rs 45
c) Rs 87, Rs 78
d) Rs 35, Rs 53

B

Question. When 2 unbiased coins are tossed once, the variance of the number of head is
a) 1
b) 3/2
c) 1/4
d) None of these

D

Question. For any random variable for which E(x) exists find the value of μ0
a) 0
b) -1
c) 2
d) 1

D

Question. If the probability density is given by Where k is appropriate constant the probability density of the random variable Y = 2X / 1 + 2X
a) g(y) = k/16y3 .(1-y) for 0 > y > 1 and g(y) = 0 elsewhere
b) g(y) = k/16y3 .(1-y) for 0 < y < 1 and g(y) = 0 elsewhere
c) g(y) = k/16y2 .(1-y) for 0 < y < 1 and g(y) = 0 elsewhere
d) g(y) = k/16y9 .(1-y) for 0 < y < 1 and g(y) = 1 elsewhere

B

Question. Two dices are thrown simultaneously and ‘getting a number less than 3’ on a die is termed as a success. Obtain the probability distribution of the number of successes
a) x/p(x), 0/(4/9), 1/(5/9), 2/(1/9)
b) x/p(x), 0/(1/9), 1/(4/9), 2/(1/9)
c) x/p(x), 0/(4/9), 1/(4/9), 2/(1/9)
d) x/p(x), 0/(4/9), 1/(7/9), 2/(1/9)

C

Question. Find variance for the random variable x that has the probability density

a) 1/9
b) 2/9
c) 4/9
d) 5/9

B

Question. If X is the number of head obtained in 4 tosses of a balanced coin then find the probability distribution of the random variable Z = (X-2)2
a) z/h(z), 0/(3/8), 1/(4/8), 4/(1/8)
b) z/h(z), 0/(1/8), 1/(4/8), 4/(1/8)
c) z/h(z), 0/(3/8), 1/(2/8), 4/(1/8)
d) z/h(z), 0/(3/8), 1/(7/8), 4/(1/8)

A

Question. If the joint density of X1 and X2 is given by

Find the probability density of Y = X1+ X2
a) f(y) = 6(ey – e-3y ) for y < 0 elsewhere f(y) = 0
b) f(y) = 6(e-2y – e-3y ) for y > 0 elsewhere f(y) = 0
c) f(y) = 6(e-2y – e-y ) for y > 0 elsewhere f(y) = 1
d) f(y) = 6(e-2y – e-y/2 ) for y > 0 elsewhere f(y) = 0

B

Question. Find μ1`of the discrete random variable x that has the probability distribution f(x) = 2(1/3x) for x = 1, 2, 3 – – –
a) 1/2
b) 0
c) 1
d) 3/2

D

Question. If the joint probability density of X and Y is given by

Find the expected value of g(X,Y) = X/Y3
a) 13/84
b) 15/84
c) 84/13
d) 84/15

B

Question. If the probability density of Xs given by

Find the expected value of g(X) = X2-5X+3
a) 11/3
b) -11/3
c) -11/6
d) 11/6

C

Question. The expected value of X is usually written as:
a) E(X) or Σ
b) E(X) or μ
c) E(X) or φ
d) E(X) or λ

B

Question. The probability distribution for

The variance of the random variable x is
a) 20
b) 21
c) 22
d) 23

A

Question. Suppose an insurance company offers a 45 year old man a Rs1,000. 1 year term insurance policy for an annual premium of Rs12 . Assuming that the number of deaths per 1000 is 5 for persons in this age this group. The expected gain for the insurance company on a policy of this type is
a) 7
b) 8
c) 9
d) 10

A

Question. In a random throw of n dice, the expectation of the sum of points on them is
a) n/2
b) 3n/2
c) 7n/2
d) 9n/2

C

Question. The joint probability density function

Of 2 random variables X and Y, find P[(X,Y)€A] where A is the region (x,y)/0 < x, ½, 1<y<2
a) 11/65
b) 11/80
c) 10/76
d) 67/80

B

Question. E(x2) = 91/6. Find the value of E(2 x2+1) is
a) 92/3
b) 91/3
c) 90/3
d) 94/3

D

Question. If X has the probability density

Find k and P(0.5 ≤ X ≤ 1)
a) 0.173
b) 0.5
c) 0.11
d) None of these

A

Question. A dice is tossed twice ‘getting a number less than 3’ is termed as success. Hence the mean of the number of successes is
a) 1
b) 3/2
c) 1/4
d) 2/3

D

Question. The moment-generating function of a random variable which has probability density f(x) = 1/2e-|x| for – ∞ < x < ∞ is
a) Mx (t) = 1/2t+1
b) Mx (t) = 1/1-t2
c) Mx (t) = 1/-2t
d) Mx (t) = 1/t2

B

Question. Find the E(X) whose probability density is given by

a) 35/12
b) 38/12
c) 37/12
d) 33/12

C

Question. If X has the probability density

Find the expected value of g(X) = e3x/4
a) 1
b) 2
c) 3
d) 4