Test: Theory of Random Variable & Noise - Electronics and Communication Engineering (ECE) MCQ

# Test: Theory of Random Variable & Noise - Electronics and Communication Engineering (ECE) MCQ

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## 10 Questions MCQ Test GATE ECE (Electronics) Mock Test Series 2025 - Test: Theory of Random Variable & Noise

Test: Theory of Random Variable & Noise for Electronics and Communication Engineering (ECE) 2024 is part of GATE ECE (Electronics) Mock Test Series 2025 preparation. The Test: Theory of Random Variable & Noise questions and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus.The Test: Theory of Random Variable & Noise MCQs are made for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Theory of Random Variable & Noise below.
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Test: Theory of Random Variable & Noise - Question 1

### A certain random variable has the CDF (cumulative distribution function) given by: Which of the following options is/are not correct?

Detailed Solution for Test: Theory of Random Variable & Noise - Question 1

Since
Now,

Also,

Hence, option (c) is not correct.

Test: Theory of Random Variable & Noise - Question 2

### A radio receiver with 10 kHz bandwidth has a noise figure of 30 dB. The noise power at input is (take K = 1.38 x 10-23 J/Kelvin and assume room temperature of 300 K)

Detailed Solution for Test: Theory of Random Variable & Noise - Question 2

Noise figure,

Test: Theory of Random Variable & Noise - Question 3

### The probability density function of a random variable is given by The value of constant K is

Detailed Solution for Test: Theory of Random Variable & Noise - Question 3

Since the area under the probability density function is equal to 1, therefore.

or,
or,

Test: Theory of Random Variable & Noise - Question 4

Consider the following statements associated with different types of noise occurring in nature:
1. The flicker noise occurs at high operating frequencies (few MHz).
2. Shot noise is produced due to random variations in the arrival of electrons at the output electrode of an amplifying device.
3. The thermal noise is proportional to temperature and the value of the resistance and is independent of bandwidth.
4. Noise can also be represented by fictitious temperature called noise temperature.
Which of the statements given above is/are not correct?

Detailed Solution for Test: Theory of Random Variable & Noise - Question 4
• The flicker noise occurs at low operating frequencies (few KHz). Hence, statement-1 is not correct.
• The thermal noise voltage is given by,
Hence, it is proportional to temperature, bandwidth and the value of resistance. Thus, statement-3 is not correct.
• Statements 1 and 4 are correct.
Test: Theory of Random Variable & Noise - Question 5

Over a certain binary communication channel (BCC), symbol ‘O' is transmitted with a probability of 0.4 and symbol ‘1’ is transmitted with a probability of 0.6. The probability of error when '1' is transmitted is 10-4 and that when a ‘0' is transmitted is 10-6.
What is the probability of error in the channel?

Detailed Solution for Test: Theory of Random Variable & Noise - Question 5

BCC with different probabilities:

According to question,

Here,  P(Y0/X0) =  Probability of error when ‘1 ’ is transmitted
P(Y1/X0) = Probability of error when ‘0 ’ is transmitted
∴ Probability of error in channel

or, P(E) = 0.604 x 10-4

Test: Theory of Random Variable & Noise - Question 6

Thermal noise is also known as

Detailed Solution for Test: Theory of Random Variable & Noise - Question 6

The electronic components generate thermal, shot noise. The thermal noise is also called Johnson noise.

Test: Theory of Random Variable & Noise - Question 7

The joint PDF of the.random variable x and y is given by,

where, a and b are constants. The value of C is

Detailed Solution for Test: Theory of Random Variable & Noise - Question 7

The joint CDF (cumulative distribution function ) is
which will be equal to 1 because the probability of  covers all the possible values of random variables X and Y.
∴
According to the definition of CDF, we have

Substituting the value of PDF, we have:

or,
or,
or,
or,

Test: Theory of Random Variable & Noise - Question 8

Assertion (A): if the random process x(t) changes very rapidly with time, the autocorrelation function will decrease rapidly to zero.
Reason (R): The autocorrelation function Rx(τ) is a measure of interdependence of two random variables obtained by observing the random process x(t) at time τ seconds apart.

Detailed Solution for Test: Theory of Random Variable & Noise - Question 8

From above figure we can conclude that both assertion and reason are true and reason is the correct explanation of assertion.

Test: Theory of Random Variable & Noise - Question 9

Consider the following statements associated with conditional probability density function (PDF):
1. The conditional PDF is basically a probability, density function (PDF).
2. The conditional PDF is a non-negative function.
3. The area under a conditional PDF is always equal to 1,
4. The conditional density functions reduce to the marginal density functions.
Which of the statements given above are correct?

Detailed Solution for Test: Theory of Random Variable & Noise - Question 9

• The conditional PDF is basically a PDF Since i.e. the conditional PDF is a non ­negative function.
Hence, statements 1 and 2 are correct.
th e area under a conditional PDF is always equal to 1. Hence, statement-3 is also correct.
• If the random variables x and y are statistically independent, then
This means that the conditional density functions reduce to the marginal density functions. Hence, statement-4 is also correct.

Test: Theory of Random Variable & Noise - Question 10

The distribution function of random variable is

Detailed Solution for Test: Theory of Random Variable & Noise - Question 10

The distribution function of a random variable is the probability that the value taken by the random variable is less than or equal to the real number x.

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