ENPR300 Probability & Random Process Assignment 04, 2025-26 | University of Dubai

ENPR300 Assignment 04

Course Code: ENPR 300           
Course Title: Probability & Random Process
Semester: Fall           
Academic year : 2025/2026
Assessment Instrument: Assignment04       
Date: November 27, 2025
Course Instructor: Dr. Diana Dawoud     
College/Department: CEIT/ Electrical Engineering

Learning Outcomes

This instrument assesses the following Course Learning Outcomes (CLO):

* Linkages to Program Outcomes and Concentration Outcomes are provided in the Syllabus.

Exam Instructions

• Answer all questions
• Time allowed is One Week.

Question 1 [22 points]

(1) Let {X_i}^∞_i=1 be a sequence of i.i.d. Bernoulli random variables with

P(X_i = 1) = 0.3,    P(X_i = 0) = 0.7.

Define the sample mean

X̄_n = (1/n) Σ_i=1ⁿ X_i.

A. [3 points] Find E[X_i] and Var(X_i).

B. [3 points] According to the Law of Large Numbers, what happens to X̄_n as n → ∞? Be precise.

C. [4 points] Suppose you observe X̄₅₀ = 0.46 for n = 50 trials. Comment on whether this contradicts the LLN. Justify your answer.

(2) The lifetime (in hours) of a certain type of electronic component has mean μ = 1200 hours and standard deviation σ = 300 hours. Assume lifetimes of different components are independent and identically distributed.

A. [4 points] Let X̄₂₅ be the sample mean lifetime of a random sample of n = 25 components. Using the Central Limit Theorem, approximate

P(X̄₂₅ > 1300).

B. [4 points] For a sample of n = 100 components, approximate

P(1150 < X̄₁₀₀ < 1250).

C. [4 points] Comment on how increasing the sample size from n = 25 to n = 100 affects the variability of the sample mean and the accuracy of the CLT approximation.

Question 2 [17 points]

(1) In a baseband communication receiver, the input noise is modeled as an ideal white Gaussian noise process with two–sided PSD

S_X(f) = N₀/2    for all f ∈ ℝ.

The receiver front–end includes an ideal lowpass filter with bandwidth B Hz and unity gain in the passband.

A. [3 points] Derive the output noise PSD S_Y(f) at the filter output.

B. [3 points] Compute the output noise power (i.e., variance) σ_Y² in terms of N₀ and B.

C. [3 points] Explain qualitatively how changing the bandwidth B affects the noise power at the receiver input, and discuss the engineering trade–off between noise reduction and signal distortion.

(2) Consider the following three random processes used to model different types of signals in a communication system:

1. X₁(t) with

E[X₁(t)] = 0,    R_X₁(t₁, t₂) = σ² e^{−|t₁ − t₂|}.

2. X₂(t) with

E[X₂(t)] = A cos(2πf₀t),    R_X₂(t₁, t₂) = σ² e^{−|t₁ − t₂|}.

3. X₃(t) with

E[X₃(t)] = 0,    R_X₃(t₁, t₂) = σ² e^−|t₁| e^−|t₂|.

A. [6 points] For each process, decide whether it is wide-sense stationary (WSS). Justify your answer in terms of the mean and the autocorrelation function.

B. [2 points] From an engineering perspective, which of these processes is most appropriate to model stationary background noise? Explain briefly.