Random Variables and Distribution Functions MCQs

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 the options

Answer: 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

Answer: B

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)

Answer: 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)

Answer: A

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

Answer: 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 the options

Answer: 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)

Answer: 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

Answer: B

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

Answer: 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

Answer: 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

Answer: D

Question. Given that X has the probability distribution f(x) = 1/8(3/x) for x = 0, 1, 2 and 3, find the moment-generating function of this random variable and use it to determine μ1`and μ2 `
a) 0
b) 3/2
c) 1/2
d) 1

Answer: 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 the options

Answer: 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

Answer: 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 the options

Answer: 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

Answer: C

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)

Answer: 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

Answer: B

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

Answer: 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

Answer: 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

Answer: 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

Answer: 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

Answer: B

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

Answer: B

Question. If the probability density of X is given by

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

Answer: 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

Answer: B

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

Answer: 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

Answer: D

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

Answer: 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

Answer: C

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

Answer: 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

Answer: B

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

Answer: 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 the options

Answer: A

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

Answer: B

Question. The probability distribution for

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

Answer: A

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 the options

Answer: D

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

Answer: D