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

MCQ on Probability & Random Signal Theory

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

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

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

Answer: 8

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

E(X²) – E²(X)
E(X²) + E²(X)
E(X²)
E²(X)

Answer: E(X²) – E²(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?

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?

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

MCQ on Probability & Random Signal Theory

Multiple-Choice Questions

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