Probability MCQ

1. For the function f(x) = a + bx, 0 ≤ x ≤ 1, to be a valid probability density function, which one of the following statements is correct?

  1. a = 0.5, b = 1
  2. a = 1, b = 4
  3. a = 1, b = -1
  4. a =0, b = 1
Answer
Answer. a

2. If a random variable X has a Poisson distribution with mean 5, then the expectation E[(X + 2)2] equals

  1. 56
  2. 54
  3. 58
  4. 60
Answer
Answer. b

3. Assume that in a traffic junction, the cycle of the traffic signal lights is 2 minutes of green (vehicle does not stop) and 3 minutes of red (vehicle stops). Consider that the arrival time of vehicles at the junction is uniformly distributed over 5 minute cycle. The expected waiting time (in minutes) for the vehicle at the junction is

  1. 0.8
  2. 0.5
  3. 0.3
  4. 0.9
Answer
Answer. d

4. The area (in percentage) under standard normal distribution curve of random variable Z within limits from-3 to +3 is

  1. 99.74
  2. 98.24
  3. 95
  4. 92.60
Answer
Answer. a

5. Let the probability density function of a random variable, X, be given as:

f_x(x)=\frac{3}{2}e^{-3x}u(x)+ae^{4x}u(-x)

where u(x) is the unit step function. Then the value of ‘a’ and Prob {X ≤ 0}, respectively, are

  1. 2, 1/2
  2. 4, 1/2
  3. 2, 1/4
  4. 4, 1/4
Answer
Answer. a

6. Probability density function of a random variable X is given below

f(x)=\begin{cases} 0.25 & \text{ if } 1\leq x\geq 5 \\ 0 & \text{} otherwise \end{cases} P(X ≤ 4)

  1. 3/4
  2. 1/2
  3. 1/4
  4. 1/8
Answer
Answer. a

7. Two random variables X and Y are distributed according to

f_{X,Y}(x,y)=\begin{cases} (x+y) & \text{} 0\leq x\leq 1, 0\leq y\leq 1 \\ 0 & \text{} otherwise \end{cases}

The probability P(X+Y ≤ 1) is

  1. 0.66
  2. 0.33
  3. 0.5
  4. 0.1
Answer
Answer. b

8. A probability density function on the interval [a, 1] is given by 1/x2 and outside this interval, the value of the function is zero. The value of a is

  1. 0.2
  2. 0.6
  3. 0.4
  4. 0.5
Answer
Answer. d

9. A random variable X has probability density function f(x) as given below

f(x)=\begin{cases} a+bx & \text{for\: } 0< x< 1 \\ 0 & \text{} otherwise \end{cases}

If the expected value E[X] = 2/3, then Pr[X < 0.5] is

  1. 1/4
  2. 3/4
  3. 1/2
  4. 3/2
Answer
Answer. a

10. The probability density function of a random variable, x is

f(x)=\frac{x}{4}(4-x^{2})\; for\; 0\leq x\leq 2 and otherwise 0.

The mean, μx of the random variable is

Answer
Answer. 1.066

11. Consider the following probability mass function (p.m.f.) of a random variable X.

p(X,q)=\begin{cases} q & \text{ if } X=0 \\ 1-q & \text{ if } X=1 \\ 0 & \text{} otherwise \end{cases}

If q = 0.4, the variance of X is

Answer
Answer. 0.24

12. Suppose Xi, for i = 1, 2, 3 are independent and identically distributed random variables whose

probability mass functions are

Pr[Xi = 0] = Pr[Xi = 1] = 1/2 for i = 1, 2, 3

Define another random variable Y = X1X2 ⊕ X3, where ⊕ denotes XOR. Then Pr[Y = 0 | X3 = 0] =

Answer
Answer. 0.75

13. Let X be a random variable with probability density function

f(x)=\begin{cases} 0.2 & \text{ for } \left | x \right |\leq 1 \\ 0.1 & \text{ for } 1< \left | x \right |\leq 4 \\ 0 & \text{} otherwise \end{cases}

The probability P(0.5 < X < 5) is

Answer
Answer. 0.4

14. The probability density function of evaporation E on any day during a year in a watershed is given by

f(E)=\begin{cases} \frac{1}{5} & \text{} 0\leq E\leq 5\; mm/day \\ 0 & \text{} otherwise \end{cases}

The probability that E lies in between 2 and 4 mm/day in a day in the watershed is (in decimal)

Answer
Answer. 0.4

15. In the following table, x is a discrete random variable and p(x) is the probability density. The standard deviation of x is

x 1 2 3
p(x) 0.3 0.6 0.1
  1. 0.18
  2. 0.36
  3. 0.54
  4. 0.6
Answer
Answer. d

16. Let X be a normal random variable with mean 1 and variance 4. The probability P{X <0} is

  1. 0.5
  2. greater than zero and less than 0.5
  3. greater than 0.5 and less than 1.0
  4. 1.0
Answer
Answer. b

17. A continuous random variable X has a probability density f(x) = e-x, 0 < x < ∞. Then P{X > 1} is

  1. 0.368
  2. 0.5
  3. 0.632
  4. 1
Answer
Answer. a

18. Find the value of λ such that function f(x) is valid probability density function

f(x)=\begin{cases} \lambda (x-1)(2-x) & \text{} 1\leq x\leq 2 \\ 0 & \text{} otherwise \end{cases}

  1. 2
  2. 4
  3. 6
  4. 8
Answer
Answer. c

19. Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. The probability that max[X, Y] is less than 1/2 is

  1. 3/4
  2. 9/16
  3. 1/4
  4. 2/3
Answer
Answer. b

20. Consider a random variable X that takes values +1 and -1 with probability 0.5 each. The values of the cumulative distribution function F(x) at x = – 1 and +1 are

  1. 0 and 1
  2. 0 and 0.5
  3. 0.25 and 0.75
  4. 0.5 and 1
Answer
Answer. d

21. If the difference between the expectation of the square of a random variable (E[x2]) and the square of the expectation of the random variable (E[x])2 is denoted by R, then

  1. R = 0
  2. R < 0
  3. R ≥ 0
  4. R > 0
Answer
Answer. c

22. Let X be a randon variable following normal distribution with mean +1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown. If P(X ≤ -1) = P(Y ≥ 2) the standard deviation of Y is

  1. 3
  2. 2
  3. 1.414
  4. 1
Answer
Answer. a

23. If probability density function of a random variable x is

f(x) = x2 for -1 ≤ x ≤ 1, and

f(x) = 0 for any other value of x

them, the percentage probability P\left ( \frac{-1}{3}\leq x\leq \frac{1}{3} \right ) is

  1. 0.247
  2. 2.47
  3. 24.7
  4. 247
Answer
Answer. b

24. Let X and Y be two independent random variables. Which one of the relations between expectation (E), variance (Var) and covariance (Cov) given below is FALSE?

  1. E(XY) = E(X) E(Y)
  2. Cov (X, Y) = 0
  3. Var (X+ Y) = Var (X) + Var (Y)
  4. E(X2 Y2) = (E(X))2 (E(Y))2
Answer
Answer. d

25. A probability density function is of the form p(x) = Ke-α|x|, x ∈ (-∞,∞).

The value of K is

  1. 0.5
  2. 1
  3. 0.5α
  4. α
Answer
Answer. c

26. Consider the continous random variable with probability density function

f(t) = 1 + t for -1 ≤ t ≤ 0

f(t) = 1 – t for 0 ≤ t ≤ 1

The standard deviation of the random variable is

a. \frac{1}{\sqrt{3}}\\ b. \frac{1}{\sqrt{6}}\\ c. \frac{1}{3}\\ d. \frac{1}{6}

Answer
Answer. b

27. Let f(x) be the continuous probability density function of a random variable X. The probability that a < X ≤ b, is

  1. f(b-a)
  2. f(b)-f(a)
  3. \int_{a}^{b}f(x)dx
  4. \int_{a}^{b}xf(x)dx
Answer
Answer. c

28. A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0, 0), (1,0), (1, 2) and (0, 2). If p is the length of the position vector of the point, the expected value of p2 is

  1. 2/3
  2. 1
  3. 4/3
  4. 5/3
Answer
Answer. d

29. An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches -0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probablity. The sum total of the expected marks obtained by all these students is

  1. 2550
  2. 0
  3. 9375
  4. 7525
Answer
Answer. c

30. A program consists of two modules executed sequentially. Let f1(t) and f2(t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by

  1. f1(t) + f2(t)
  2. \int_{0}^{t}f_1(x)f_2(x)dx
  3. \int_{0}^{t}f_1(x)f_2(t-x)dx
  4. max{f1(t), f2(t)}
Answer
Answer. c
error: Content is protected !!