01 Graded Assignment 10

Exercise Questions ❓ Solutions ☀️ Here are the step-by-step solutions for each question from the uploaded images, covering the concepts of Bayesian inference, Maximum Likelihood Estimation (MLE), and Probability Distributions. Question 1 Problem: Outcomes on rolling a die ten times are: 6, 4, 3, 6, 1, 5, 4, 6, 4, 2. Use the Uniform[0, 1] prior to find the posterior mean of $p$, which denotes the probability of getting an even number. Write your answer correct to two decimal places.

02 Graded Assignment 11

Exercise Questions ❓ Solutions 🟩 Here are the detailed solutions and concept explanations for the statistical problems provided in the images. Question 1 Problem: The average marks scored by students… is reported to be 400 with a standard deviation of 5. You suspect the average may be lower, possibly 390… What sample size do you need for a test at significance level 0.05 and power 0.95?

03 Graded Assignment 12

Exercise Questions ❓ SOlutions 🟩 Here are the detailed solutions for the statistical problems. Question 1 Problem: Comparing IQs of two batches. Batch 1: $n_1=25, \bar{x}_1 = 110, \sigma_1 = 8$ Batch 2: $n_2=25, \bar{x}_2 = 115, \sigma_2 = 7$ Test for significant difference at $\alpha = 0.05$. Concept: Two-Sample Z-Test for Means When comparing the means of two independent populations with known standard deviations (or large samples), we use the Z-statistic:

04 Graded Assignment 2

Exercise Questions 🔥 Exercise Solutions 🧯 Question 1: Geometric Distribution & Memoryless Property (from file image_a508e1.png) The Question: A software engineer is testing a program for bugs. Each run of the program has a 15% chance of encountering a bug. Let X be a geometric random variable representing the number of runs until the first bug is found. Given that the program has already been run 3 times without finding any bugs, what is the probability that the first bug will be found after the 5th run?

05 Graded Assignment 3

Exercise Questions 🧯 Exercise Solutions 🤘 Good morning! Here in India on this Wednesday, let’s explore this excellent collection of questions covering a range of important topics in probability and statistics. Questions 1 & 2: Markov’s and Chebyshev’s Inequalities (from file image_996640.png) Common Data: The number of people ($X$) who make a reservation in a restaurant on a particular day is a random variable with mean $E(X) = \mu = 10$ and variance $Var(X) = \sigma^2 = 5$.

06 Graded Assignment 4

Exercise Questions ❓ Exercise Solutions 🧪 Good morning! Here in India on this Monday, let’s explore these questions. They cover key concepts related to continuous random variables, including Probability Density Functions (PDFs), Cumulative Distribution Functions (CDFs), percentiles, conditional probabilities, and applications of the Normal distribution. Core Concepts: Continuous Random Variables Probability Density Function (PDF), $f_X(x)$:

07 Graded Assignment 5

Exercise Questions ❓ Exercise Solutions 🔬 Hello! I can certainly help you solve these problems. Here is a detailed breakdown of each question, including the core concepts and a step-by-step solution. ❓ Question 1: Battery Life A person randomly chooses a battery from a store which has 90 batteries of type A and 260 batteries of type B. Battery life of type A and type B batteries are exponentially distributed with average life of 9.0 years and 13.0 years, respectively. If the chosen battery lasts for 5 years, what is the probability that the battery is of type A? Enter your answer correct to two decimals accuracy.

08 Graded Assignment 6

Exercise Questions ❓ Exercise Solutions 📖 Of course. Here are the detailed solutions and conceptual explanations for all 10 problems from the images you provided. 1. Uniform Distribution on a Circle Question: Let $(X, Y) \sim \text{Uniform}(D)$, where $D = {(x, y) : (x - 3)^2 + (y - 3)^2 \le 1}$. Calculate $P(X > Y)$. Enter your answer correct to two decimals accuracy.

09 Graded Assignment 7

Exercise Questions ❓ Solutions Here are the detailed answers and conceptual explanations for each of the questions you provided. ❓ Question 1: Comparing Variances Let $X_1, X_2, X_3$ are three independent and identically distributed random variables with mean $\mu$ and variance $\sigma^2$. Given below are 3 different formulations of sample mean. (Observe that $E[A] = E[B] = E[C]$). $A = \frac{X_1 + X_2 + X_3}{3}$ $B = 0.1X_1 + 0.3X_2 + 0.6X_3$ $C = 0.2X_1 + 0.3X_2 + 0.5X_3$ Choose the correct option from the following:

10 Graded Assignment 8

Exercise Questions ❓ Exercise Solutions 🟩 Here are the detailed solutions and concept explanations for each question you provided. ❓ Question 1 Problem: Let the moment generating function (MGF) of a random variable $X$ be given by:

11 Graded Assignment 9

Exercise Questions ❓ Exercise Solutions 🟩 Hello Aryan. It looks like you are working through some solid Method of Moments (MME) and Maximum Likelihood Estimation (MLE) problems for your Statistics II course. These are foundational concepts for inference. Here are the step-by-step solutions and concept explanations for the 10 questions extracted from the images, formatted exactly as you requested.

12 Systems of Linear Equations

13 🗓️ Mock for Quiz 2

Exercise Questions ❓ Solutions 🟩 Here are the detailed solutions and concept explanations for the next set of questions. ❓ Question 1 Question: Suppose $X_1, X_2, \dots, X_{10} \sim iid \text{ Exp}(5)$. Compute the expected value and variance of sample mean, $\bar{X} = \frac{X_1 + \dots + X_{10}}{10}$.

14 🗓️ Mock for Week 5 and 6

Exercise Questions ❓ Exercise Solutions 🟩 Here are the detailed solutions and concept explanations for each of the questions you provided. ❓ Question 1 (Using information from the image with questions 1, 2, and 3) Question: Let the random variables $X$ and $Y$ have the joint density function: $f_{XY}(x, y) = \begin{cases} 1 & \text{for } 0 \le x < 1, 0 \le y < 1 \ 0 & \text{otherwise} \end{cases}$

15 Joint PMF of two discrete random variables

16 Statistics II

IIT Madras has launched the BS in Data Science and Applications. In this program, the course contents are delivered online and can be studied by anyone from anywhere, while the monthly quizzes and final semester exams will have to be attended in-person at designated centres.

17 Graded Assignment 1

Of course! Here are the solutions and explanations for all the questions you provided. Question 1: Conditional Probability Problem: The joint PMF of two discrete random variables $X$ and $Y$ is given by: $f_{XY}(x, y) = \begin{cases} \frac{1}{36}(3x + y), & x, y \in {0, 1, 2} \ 0, & \text{otherwise} \end{cases}$ What is the value of $P(0 < X < 2 \ | \ Y > 1)$? Enter the answer correct to two decimal places.