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?

- a = 0.5, b = 1
- a = 1, b = 4
- a = 1, b = -1
- 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

- 56
- 54
- 58
- 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

- 0.8
- 0.5
- 0.3
- 0.9

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

- 99.74
- 98.24
- 95
- 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

- 2, 1/2
- 4, 1/2
- 2, 1/4
- 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)

- 3/4
- 1/2
- 1/4
- 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

- 0.66
- 0.33
- 0.5
- 0.1

8. A probability density function on the interval [a, 1] is given by 1/x^{2} and outside this interval, the value of the function is zero. The value of a is

- 0.2
- 0.6
- 0.4
- 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

- 1/4
- 3/4
- 1/2
- 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 = X_{1}X_{2} ⊕ X_{3}, where ⊕ denotes XOR. Then Pr[Y = 0 | X_{3} = 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 |

- 0.18
- 0.36
- 0.54
- 0.6

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

- 0.5
- greater than zero and less than 0.5
- greater than 0.5 and less than 1.0
- 1.0

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

- 0.368
- 0.5
- 0.632
- 1

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

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

- 2
- 4
- 6
- 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

- 3/4
- 9/16
- 1/4
- 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

- 0 and 1
- 0 and 0.5
- 0.25 and 0.75
- 0.5 and 1

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

- R = 0
- R < 0
- R ≥ 0
- 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

- 3
- 2
- 1.414
- 1

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

f(x) = x^{2} 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

- 0.247
- 2.47
- 24.7
- 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?

- E(XY) = E(X) E(Y)
- Cov (X, Y) = 0
- Var (X+ Y) = Var (X) + Var (Y)
- E(X
^{2}Y^{2}) = (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

- 0.5
- 1
- 0.5α
- 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

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

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 p^{2} is

- 2/3
- 1
- 4/3
- 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

- 2550
- 0
- 9375
- 7525

30. A program consists of two modules executed sequentially. Let f_{1}(t) and f_{2}(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

- f
_{1}(t) + f_{2}(t) - $\int_{0}^{t}f_1(x)f_2(x)dx$
- $\int_{0}^{t}f_1(x)f_2(t-x)dx$
- max{f
_{1}(t), f_{2}(t)}