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

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

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

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

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

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

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

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

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

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, μ_x of the random variable is

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

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

probability mass functions are

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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. $\int_{a}^{b}xf(x)dx$

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

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

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)}