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Let the random variable X have the probability distribution listed in the table below. Determine the probability distributions of the random variable (X+1)². k 2 3 4 5 6 Pr(X=k) 0.2 0.2 0.2 0.3 0.1 Fill in the table for the probability distribution of the variable (X+1)². Pr((x + 1)² = k) k 9 16 25 36 49

Let the random variable X have the probability distribution listed in the table below. Determine the probability distributions of the random variable (X+1)². k 2 3 4 5 6 Pr(X=k) 0.2 0.2 0.2 0.3 0.1 Fill in the table for the probability distribution of the variable (X+1)². Pr((x + 1)² = k) k 9 16 25 36 49

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Title: Understanding Probability Distribution Transformation** **Objective:** Learn how to determine the probability distribution of a transformed random variable. **Problem Statement:** Let the random variable \( X \) have the probability distribution listed in the table below. Determine the probability distribution of the random variable \((X + 1)^2\). **Given Probability Distribution:** | \( k \) | \( \Pr(X = k) \) | |---------|-----------------| | 2 | 0.2 | | 3 | 0.2 | | 4 | 0.2 | | 5 | 0.3 | | 6 | 0.1 | **Task:** Fill in the table for the probability distribution of the variable \((X + 1)^2\). **Transformations:** - When \( X = 2 \), \((X + 1)^2 = 3^2 = 9\). - When \( X = 3 \), \((X + 1)^2 = 4^2 = 16\). - When \( X = 4 \), \((X + 1)^2 = 5^2 = 25\). - When \( X = 5 \), \((X + 1)^2 = 6^2 = 36\). - When \( X = 6 \), \((X + 1)^2 = 7^2 = 49\). **Table for \((X + 1)^2\):** | \( k \) | \( \Pr((X + 1)^2 = k) \) | |---------|-------------------------| | 9 | | | 16 | | | 25 | | | 36 | | | 49 | | **Instructions:** Calculate the probabilities for each transformed value and fill the table. The probability for each value is the sum of probabilities that map to it based on the transformation. **Example Calculation for Clarity:** Determine \(\Pr((X + 1)^2 = 9)\): - Based on transformation, \(\Pr(X = 2)\) fully contributes to \(\Pr((X + 1)^2 = 9)\). - Therefore, \(\Pr((X + 1)^2 = 9) =
Transcribed Image Text:**Title: Understanding Probability Distribution Transformation** **Objective:** Learn how to determine the probability distribution of a transformed random variable. **Problem Statement:** Let the random variable \( X \) have the probability distribution listed in the table below. Determine the probability distribution of the random variable \((X + 1)^2\). **Given Probability Distribution:** | \( k \) | \( \Pr(X = k) \) | |---------|-----------------| | 2 | 0.2 | | 3 | 0.2 | | 4 | 0.2 | | 5 | 0.3 | | 6 | 0.1 | **Task:** Fill in the table for the probability distribution of the variable \((X + 1)^2\). **Transformations:** - When \( X = 2 \), \((X + 1)^2 = 3^2 = 9\). - When \( X = 3 \), \((X + 1)^2 = 4^2 = 16\). - When \( X = 4 \), \((X + 1)^2 = 5^2 = 25\). - When \( X = 5 \), \((X + 1)^2 = 6^2 = 36\). - When \( X = 6 \), \((X + 1)^2 = 7^2 = 49\). **Table for \((X + 1)^2\):** | \( k \) | \( \Pr((X + 1)^2 = k) \) | |---------|-------------------------| | 9 | | | 16 | | | 25 | | | 36 | | | 49 | | **Instructions:** Calculate the probabilities for each transformed value and fill the table. The probability for each value is the sum of probabilities that map to it based on the transformation. **Example Calculation for Clarity:** Determine \(\Pr((X + 1)^2 = 9)\): - Based on transformation, \(\Pr(X = 2)\) fully contributes to \(\Pr((X + 1)^2 = 9)\). - Therefore, \(\Pr((X + 1)^2 = 9) =
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