Question A uniformly distributed random variable X with probability density function f X (x) = (1/10) ( u(x+5) − u(x−5) ) where u(.) is the unit step function, is passed through a transformation given in the figure below. The probability density function of the transformed random variable Y would be... Y = 1 when X ∈ [−2.5, 2.5], else 0 (a) f Y (y) = (1/5)(u(y+2.5) − u(y−2.5)) (b) f Y (y) = 0.5δ(y) + 0.5δ(y−1) (c) f Y (y) = 0.25δ(y+2.5) + 0.25δ(y−2.5) + 0.5δ(y) (d) f Y (y) = 0.25δ(y+2.5) + 0.25δ(y−2.5) Correct Answer The transformation maps X to Y such that: If X ∈ [-2.5, 2.5], then Y = 1 If X < −2.5 or X...

Object vs Function vs Method (Python) In Python, these three concepts are closely related, which is why they can feel confusing at first. The key idea is who owns what . 1. Function A function is a reusable block of code that performs a task. It exists independently . def greet(name): return "Hello " + name greet("Alex") Stands alone Not tied to any object Can be called from anywhere Think: “Do this task.” 2. Object An object represents a thing. It holds data (attributes) and behavior (methods) . user = { "name": "Alex", "age": 25 } Or more commonly in Python, using a class: class User: pass u = User() Objects store related data Created from classes Almost everything i...

Radix Sort Algorithm The Radix Sort algorithm sorts an array by individual digits, starting with the least significant digit (the rightmost one). What Radix Sort Is Radix sort sorts numbers digit by digit , starting from the least significant digit (units place) , then tens, hundreds, etc. It does NOT compare numbers directly like bubble sort or quick sort. Key Mathematical Idea Extracting a digit This line is the heart of radix sort: radixIndex = (val // exp) % 10 exp is a place-value divisor Breakdown (pure math): Expression Meaning val // exp Shifts digits to the right % 10 Extracts the last digit...

Important Python Collection Data Types Python provides several built-in collection data types. The most commonly used are list , tuple , set , and dictionary . 1. List Description: Ordered and mutable collection Allows duplicate values Mutable (can be changed) Uses square brackets [] lst = [1, 2, 3, 3] lst[0] = 10 When to use: When data needs to be ordered and modified frequently. 2. Tuple Description: Ordered and immutable collection Allows duplicate values Cannot be changed after creation Uses parentheses () t = (1, 2, 3) # t[0] = 10 ❌ Not allowed When to use: For fixed data such as coordinates or records. 3. Set Description: Unordered collection of uniq...

Mean Square Value (MSV) (or Average Power) Simulator Analog (Continuous) Digital (Discrete) Analog Random Variable PDF Type Uniform Triangular Gaussian (Approx.) Custom PDF Custom f(x) (in terms of x) Lower Limit (a) Upper Limit (b) Calculate MSV Digital Random Variable Enter values and probabilities (comma separated) Values (xᵢ) Probabilities (pᵢ) Calculate MSV

Question A random variable X with uniform density in the interval 0 ≤ X ≤ 1 is quantized as follows: If 0 ≤ X ≤ 0.3, X q = 0 If 0.3 < X ≤ 1, X q = 0.7 where X q is the quantized value of X . The root mean square value of the quantization noise is: (a) 0.573 (b) 0.198 (c) 2.205 (d) 0.266 Solution Given Random variable X ∼ Uniform(0,1) Non-uniform quantizer: X q = { 0, 0 ≤ X ≤ 0.3 0.7, 0.3 < X ≤ 1 } Quantization error: e = X − X q Required: e rms = √E[e²] Step 1: PDF of X ...

Dijkstra’s Algorithm Explained Dijkstra’s algorithm is used to find the minimum cost path from a single source node to all other nodes in a graph with non-negative edge weights . Unlike Floyd–Warshall (all pairs), Dijkstra focuses on one starting point . For Example Each letter ( a – z ) is a node Each allowed transformation is a directed edge Each edge has a cost We want the cheapest cost from a source letter to all others 1. Think of Letters as Cities Imagine each letter is a city and each transformation is a one-way road with a toll. a → b (2) a → c (4) b → c (1) c → d (3) You start in city a and want to find the cheapest way to reach every other city. 2. Distance Array We ...

SSH in the terminal means Secure Shell . It is a protocol used to securely connect from one computer to another over a network , most commonly to control a remote server through a command-line interface (the terminal). What it’s used for Logging into remote servers Running commands on another machine Managing servers and cloud instances Transferring files securely (via SCP or SFTP) Basic idea You type an SSH command in your terminal, and it opens a secure, encrypted session to another computer. Example ssh user@server_address user = username on the remote machine server_address = IP address or domain name of the remote machine Why it matters Encrypted (safe from eavesdropping) Replaces insecure methods like Telnet Standard tool in Linux, macOS, and developer workflows Step-by-step example of using SSH Step 1: Open a terminal Linux / macOS : Open Terminal ...