## MCQ on Probability & Random Signal Theory

*MCQ on Probability & Random Signal Theory*, *Objective Questions on Probability & Random Signal Theory*, Gate questions on Probability & Random Signal Theory, Probability MCQ, Random Signal Theory MCQ, Probability & Random Signal Theory MCQ

### Multiple-Choice Questions

1. Which of the following is incorrect?

- A-B=A\bar{B}
- \bar{AB}=\bar{A}\bar{B}
- A.\bar{A}=0
- A.A=A

**Answer:** \bar{AB}=\bar{A}\bar{B}

2. Pick the odd one out among

- stochastic variable
- stochastic function
- random variable
- random experiment

**Answer:** random experiment

3. Pick the odd one out among

- binomial distribution
- normal distribution
- uniform distribution
- Rayleigh distribution

**Answer:** binomial distribution

4. Which of the following is incorrect?

*P(S)= 1*- P(\bar{A})=P(A)-1
- 0\leq P(A)\leq 1
- If A and B are mutually exclusive, then
*P(A + B) = P(A) + P(B)*

**Answer:** P(\bar{A})=P(A)-1

5. The total area under the probability distribution curve is

- 1
- 0
- depends on the nature of the distribution
- none of the above

**Answer:** 1

6. The spectral density of white noise

- varies with frequency
- varies with bandwidth
- varies with the amplitude of the signal
- is constant

**Answer:** is constant

7. The theoretical power of white noise is

- zero
- finite
- infinite
- depends on the frequency of the signal

**Answer:** depends on the frequency of the signal

8. The stationary process has

- ensemble average equal to time average
- all the statistical properties dependent on time
- all the statistical properties independent of time
- zero variance

**Answer:** all the statistical properties independent of time

9. Events A and B are statistically independent if

- A and B occur simultaneously
- A and B occur at different times
- occurrence of A includes occurrence of B
- none of the above

**Answer:** none of the above

10. Pick the odd man out among

- expectation
- variance
- standard deviation
- Tchebycheff’s inequality

**Answer:** Tchebycheff’s inequality

11. The probability density function of a random variable X is ae^{-bx}u(x). Then

- a and b can be arbitrary
- a = b/2
- a = b
- a = 2b

**Answer:** a = b/2

12. The density function of a random variable X is given by: f(x)=\left\{\begin{matrix} \frac{1}{b-a}, & a\leq x\leq b\\ 0, & otherwise \end{matrix}\right.

The variable X is said to have

- Poisson distribution
- Gaussian distribution
- Rayleigh distribution
- uniform distribution

**Answer:** uniform distribution

13. The probability density function of a random variable is given by p(x)=ke^{-x^{2}} -∞≤x≤∞. The value of k should be

- \frac{1}{\sqrt{2\pi }}
- \sqrt{\frac{2}{\pi}}
- \frac{1}{2\sqrt{\pi }}
- \frac{1}{\pi \sqrt{2}}

**Answer:** \frac{1}{\sqrt{2\pi }}

14. For a random various x with probability function p_{(x)} shown in the following figure, the mean and the variance are respectively

- \frac{1}{2}and\frac{2}{3}
- 1and\frac{4}{3}
- 1and\frac{2}{3}
- 2and\frac{4}{3}

**Answer:** 1and\frac{4}{3}

15. The spectral density of a real-valued random process has

- an even symmetry
- an odd symmetry
- a conjugate symmetry
- no symmetry

**Answer:** an even symmetry

16. If the variance \sigma _{x}^{2} of d_{n}=x_{n}-x_{n-1} is one-tenth of variance \sigma _{x}^{2} of a stationary zero-mean discrete time signal x(0, then the normalized autocorrelation function R_{XX(K)}/\sigma _{x}^{2} at K= 1 is

- 0.95
- 0.90
- 0.10
- 0.05

**Answer:** 0.95

17. Let Y and Z be the random variables obtained by sampling X_{(t)} at t = 2 and t = 4 respectively. Let W = Y- Z. The variance of W is

- 13.36
- 9.36
- 2.64
- 8.00

**Answer:** 2.64

18. A random process obeys Poisson’s distribution. It is given that the mean of the process is 5. Then the variance of the process is

- 5
- 0.5
- 25
- 0

**Answer:** 5

19. A ternary 5 source produces alphabets A, B and C with probabilities P_{A} = P_{B} = p and P_{C}. Which one of the following gives correct values for the maximum value of the entropy of the source and the corresponding value of p and the range of p?

- 1.58. 0.33, (0, 0.5)
- 1.0, 0.5, (0, 1)
- 3.0. 0.67, (0, 0.5)
- 2.0, 4.2, (0, 0.3)

**Answer:** 3.0. 0.67, (0, 0.5)

20. An output of a communication channel is a random variable *v* with the probability density function as shown in the figure. The mean square value of *v* is

- 4
- 6
- 8
- 9

**Answer:** 8

21. If E denotes expectation, the variance of a random variable X is given by

- E(X
^{2}) – E^{2}(X) - E(X
^{2}) + E^{2}(X) - E(X
^{2}) - E
^{2}(X)

**Answer:** E(X^{2}) – E^{2}(X)

22. Which one of the following is the correct statement? If the value of a resistor creating thermal noise is doubled, the noise generated is

- halved
- doubled
- unchanged
- slightly changed

**Answer:** unchanged

23. P_{X}(x)=Me^{-2|x|}+Ne^{-3|x|} is the PDF for the real random variable X over the entire x axis. M and N are both positive real numbers. The equation relating M and N is

- M+\frac{2}{3}N=1
- 2M+\frac{1}{3}N=1
- M+N=1
- M+N=3

**Answer:** M+\frac{2}{3}N=1

24. A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads?

- (\frac{1}{2})
- 10_{C_{2}}(\frac{1}{2})^{2}
- (\frac{1}{2})^{10}
- 10_{C_{2}}(\frac{1}{2})^{10}

**Answer:** (\frac{1}{2})^{10}

25. Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0, 1 and 2 with probabilities ½, ¼, and 1/4 respectively. What is the conditional probability P_{(X+Y=2/X=Y=0)}?

- 0
- 1/16
- 1/6
- 1

**Answer:** 1/6

26. A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean of X as 3.5 and her teacher calculates the variance X as 1.5. Which of the following statements is true?

K | 1 | 2 | 3 | 4 | 5 |

P_{(X=K)} | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |

- Both the student and the teacher are right.
- Both the student and the teacher are wrong.
- The student is wrong but the teacher is right.
- The student is right but the teacher is wrong.

**Answer:** Both the student and the teacher are wrong.

27. A fair coin is tossed independently four times. The probability of the event “the number of times heads show up is more than the number of times tails show up” is

- 1/16
- 1/8
- 1/4
- 5/16

**Answer:** 5/16

28. A fair dice is tossed two times. The probability that the second toss results in a value that is higher that the first toss is

- 2/36
- 2/6
- 5/12
- 1/2

**Answer:** 5/12

29. Given that f(y)=|y|/y and q is any non-zero real number, the value of [f_{(q)}-f_{(-q)}] is

- 0
- -1
- 1
- 2

**Answer:** 2

30. Three friends R, S, and T shared toffees from a bowl. R took 1/3rd of the toffees but returned four to the bowl. S took 1/4th of what was left but returned three toffees to the bowl. T took half of the remainder but returned two back in the bowl. If the bowl had 17 toffees left, how many toffees were originally there in the bowl?

- 38
- 31
- 48
- 41

**Answer:** 48

31. There are two candidates P and Q in an election. During the campaign, 40% of the voters promised to vote for P and the rest, for Q. However, on the day of the election, 15% of the voters went back on their promise to vote for P and voted for Q. 25% of the voters went back on their promise to vote for Q and instead voted for P. Suppose P lost by 2 votes, then what was the total number of voters?

- 100
- 110
- 90
- 95

**Answer:** 100

32. The classical approach for probability theory does not explain the situation when the number of outcomes of an experiment is small.

- True
- False

**Answer:** False

33. Mutually exclusive events are also statistically independent.

- True
- False

**Answer:** False

34. The probability that a continuous random variable takes on a particular value is zero.

- True
- False

**Answer:** True

35. Unit of variance is same as that of the random variable.

- True
- False

**Answer:** False