RANDOM SIGNALS AND NOISE EXAM STUDY GUIDE 2025/2026 (A. MA) Accurate Questions and Verified Correct Solutions with Detailed Rationales || 100% Guaranteed Pass Latest Version

RANDOM SIGNALS AND NOISE EXAM STUDY
GUIDE 2025/2026 (A. MA) Accurate Questions and
Verified Correct Solutions with Detailed Rationales
|| 100% Guaranteed Pass
<Latest Version>

Description: Full exam prep resource for Random Signals and Noise, focusing on
probability theory, stochastic processes, autocorrelation, and spectral density
analysis. Designed for engineering and applied mathematics students.


Keywords: random signals, noise analysis, stochastic processes, signal processing,
probability, exam 2025




Random Signals and Noise Exam Study Guide 2025/2026 (A. MA)
100 Questions and Verified Correct Solutions


Section 1: Probability Theory Fundamentals
1. What is the sample space in a random experiment?
A) The set of all possible outcomes
B) The most likely outcome
C) The average value of the outcomes
D) The set of all failed outcomes

Correct Answer: A) The set of all possible outcomes
Rationale: The sample space, often denoted by S or Ω, is a fundamental concept
defined as the collection of all distinct possible results of a random experiment.

,2. Two events A and B are independent if:
A) P(A ∩ B) = P(A) + P(B)
B) P(A | B) = P(A)
C) A and B cannot occur simultaneously
D) P(A ∪ B) = P(A)P(B)

Correct Answer: B) P(A | B) = P(A)
Rationale: Independence means the occurrence of B does not affect the probability
of A. This is mathematically defined as P(A|B) = P(A), which is equivalent to P(A
∩ B) = P(A)P(B).
3. The probability of an event A, given that event B has occurred, is calculated
as:
A) P(A|B) = P(A ∪ B) / P(B)
B) P(A|B) = P(A ∩ B) / P(B)
C) P(A|B) = P(B) / P(A ∩ B)
D) P(A|B) = P(A) / P(B)

Correct Answer: B) P(A|B) = P(A ∩ B) / P(B)
Rationale: This is the definition of conditional probability, provided P(B) > 0.
4. For a continuous random variable X, the probability that X takes on a
specific value 'a' is:
A) F_X(a) (the CDF)
B) f_X(a) (the PDF)
C) Always 0.5
D) Always 0

Correct Answer: D) Always 0
Rationale: For a continuous random variable, probability is defined over an
interval. The probability at a single point is the integral from a to a, which is zero.
5. The function that gives the probability that a random variable X is less than
or equal to a certain value x is called the:
A) Probability Density Function (PDF)
B) Cumulative Distribution Function (CDF)

,C) Moment Generating Function (MGF)
D) Probability Mass Function (PMF)

Correct Answer: B) Cumulative Distribution Function (CDF)
Rationale: By definition, the CDF, F_X(x), is F_X(x) = P(X ≤ x).
6. The expected value of a function g(X) of a continuous random variable X is:
A) ∫ g(x) f_X(x) dx
B) ∫ x f_X(x) dx
C) ∑ g(x) p_X(x)
D) ∫ g(x) F_X(x) dx

Correct Answer: A) ∫ g(x) f_X(x) dx
Rationale: This is the law of the unconscious statistician for continuous random
variables, where f_X(x) is the PDF.
7. The variance of a random variable X is defined as:
A) E[X²]
B) E[X]²
C) E[(X - E[X])²]
D) E[X] - E[X²]

Correct Answer: C) E[(X - E[X])²]
Rationale: Variance measures the average squared deviation from the mean, E[X].
8. Two random variables X and Y are uncorrelated if:
A) Their joint PDF is the product of their marginal PDFs.
B) E[XY] = 0
C) Cov(X, Y) = 0
D) They are independent.

Correct Answer: C) Cov(X, Y) = 0
Rationale: Uncorrelated specifically means their covariance is zero. Independence
(A) implies being uncorrelated, but not vice-versa.
9. The Gaussian (Normal) distribution is completely described by its:
A) Mean and Mode
B) Mean and Variance

, C) Median and Skewness
D) Variance and Kurtosis

Correct Answer: B) Mean and Variance
Rationale: The entire structure of the Gaussian PDF, f_X(x) = (1/√(2πσ²)) exp(-(x-
μ)²/(2σ²)), depends only on the parameters μ (mean) and σ² (variance).
10. The Central Limit Theorem states that the sum of a large number of
independent and identically distributed (i.i.d.) random variables:
A) Will have the same distribution as the individual variables.
B) Will follow a uniform distribution.
C) Tends to follow a Gaussian distribution, regardless of the original distribution.
D) Will have a variance of zero.

Correct Answer: C) Tends to follow a Gaussian distribution, regardless of
the original distribution.
Rationale: This is the essence of the Central Limit Theorem, making the Gaussian
distribution fundamental in statistics and signal processing.


Section 2: Random Variables and Transformations
11. If Y = aX + b, where a and b are constants, then the variance of Y is:
A) a Var(X)
B) a² Var(X)
C) |a| Var(X)
D) Var(X) + b

Correct Answer: B) a² Var(X)
Rationale: Var(aX + b) = E[(aX+b - E[aX+b])²] = E[(aX - aE[X])²] = a²E[(X -
E[X])²] = a² Var(X). The constant b shifts the mean but does not affect variance.
12. The correlation between two random variables X and Y is defined as:
A) E[X Y]
B) E[(X - E[X])(Y - E[Y])]
C) Cov(X, Y) / (σ_X σ_Y)
D) E[X] E[Y]