Problem 13 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The S.D. of a variate \(x\) is \(\sigma .\) The S.D. of the variate \(\frac{a x+b}{c}\) where \(a, b, c\) are constants, is (A) \(\left(\frac{a}{c}\right) \sigma\) (B) \(\left|\frac{a}{c}\right| \sigma\) (C) \(\left(\frac{a^{2}}{c^{2}}\right) \sigma\) (D) None of these

Step-by-Step Solution

Answer

(B) \( \left|\frac{a}{c}\right| \sigma \) is the correct option.

Step 1: Understand the Problem

We are given a variate \( x \) with standard deviation \( \sigma \). We need to find the standard deviation of a transformed variate \( \frac{a x + b}{c} \).

Step 2: Transform the Variate

The new variate \( y \) is defined as \( y = \frac{a x + b}{c} \). This can be rewritten as \( y = \frac{a}{c}x + \frac{b}{c} \).

Step 3: Use Standard Deviation Rules

Recall that the standard deviation of a constant times a variate is the absolute value of the constant times the standard deviation of the variate. Also, adding a constant \( \frac{b}{c} \) does not change the standard deviation.

Step 4: Calculate New Standard Deviation

The standard deviation of \( \frac{a}{c}x \) is \( \left|\frac{a}{c}\right| \sigma \). The constant \( \frac{b}{c} \) does not affect the standard deviation.

Step 5: Identify the Correct Option

The correct expression for the standard deviation of \( \frac{a x + b}{c} \) is \( \left|\frac{a}{c}\right| \sigma \). This matches option (B).

Key Concepts

Variate Transformation

Variate transformation is a technique used to change an existing random variable into a new one by applying a mathematical function. In the context of statistics, transforming a variate allows us to better analyze or visualize data. For instance, consider a variate \( x \) which follows a certain distribution and has a standard deviation \( \sigma \). When we transform this variate using a formula like \( \frac{a x + b}{c} \), we're effectively altering its distribution characteristics.

The process involves scaling (multiplying by \( \frac{a}{c} \)) and shifting (adding \( \frac{b}{c} \)) the original variate. It's important to note:

• Scaling affects the spread, and thus, the standard deviation of the variate. Specifically, multiplying by a constant scales the standard deviation by the absolute value of that constant.
• Shifting does not affect the spread, so the addition of a constant does not change the standard deviation of a variate.

Understanding variate transformation is crucial in probability and statistics because it allows manipulation of data to meet the specific assumptions of statistical models, such as normality, which facilitates the application of various statistical tests.

Probability and Statistics

Probability and statistics are mathematical disciplines that deal with the analysis and interpretation of random phenomena and data. They help us understand the likelihood of events and provide a framework for making informed decisions based on data.

In the world of probability, we often work with random variables, each having a probability distribution that depicts all possible outcomes and their respective probabilities. Standard deviation is a key statistical measure used in both probability and statistics to quantify the amount of variation or dispersion in a set of values.

The rule applied in the solution of this problem is grounded in the properties of standard deviation in transformed data:

• The standard deviation of a random variable multiplied by a constant \( a \) becomes \( |a| \sigma \), where \( \sigma \) is the original standard deviation.
• Adding or subtracting a constant value does not influence the standard deviation.

This is because standard deviation only measures spread and thus is unaffected by linear shifts. By understanding these fundamental concepts, students can tackle a broad range of statistical problems with greater ease.

Mathematical Problem Solving

Mathematical problem solving involves applying mathematical concepts and procedures to analyze and solve problems. In this case, the problem required understanding a conceptual transformation of a variate and calculating its new standard deviation.

Let's break down the approach:

• Identify the given information and the specific requirement of the problem. Here, the initial variate \( x \) has a standard deviation \( \sigma \).
• Transform the original variate into a new form using the specified function \( \frac{a x + b}{c} \). This introduces parameters that alter the distribution.
• Apply mathematical principles, specifically the rules of standard deviation, to the transformed variate. Recognize that multiplying by a factor influences the standard deviation, whereas adding a constant does not.
• Select the correct expression from the given options for the new standard deviation, which in this exercise is \( \left|\frac{a}{c}\right| \sigma \). This requires comprehension of why absolute values matter in standard deviation transformations.

By emphasizing these steps, students can understand not only how to solve statistical problems but also why these solutions are correct. This holistic approach develops critical thinking and analytical skills necessary for advanced studies in mathematics and statistics.