Question

If the variance of \(\mathrm{x}\) is 4 then the variance of \(3+5 \mathrm{x}\) is (a) 100 (b) 103 (c) 20 (d) 23

Short answer

The variance of the transformed random variable (3 + 5X) is 100, so the correct answer is (a) 100.

Step-by-step solution

Recall the formula for variance of a linear transformation of a random variable

The variance of a linear transformation of a random variable can be found using the following formula: \(Var(a + bX) = b^2 * Var(X)\) where a and b are constants and X is the random variable. In this case, a = 3, b = 5, and Var(X) = 4.

Plug the given values into the formula

Now, we plug the values into the formula: Var(3 + 5X) = 5^2 * Var(X)

Calculate the variance of the transformed random variable

Perform the calculation: Var(3 + 5X) = 5^2 * 4 = 25 * 4 = 100 So, the variance of the transformed random variable (3 + 5X) is 100.

Identify the correct answer

Compare the calculated value of the variance with the given options: (a) 100 (b) 103 (c) 20 (d) 23 The correct answer is (a) 100.

Key concepts

variance calculation

Variance calculation helps us understand how much a set of random variables is spread out or dispersed. Simply put, it measures the average of the squared differences from the mean. The formula for calculating variance for a random variable, represented as \[ Var(X) = E[(X - \mu)^2] \]where \( \mu \) is the mean and \( E \) denotes the expected value, serves as a foundation for understanding how data points deviate from the mean. In the context of linear transformations, the variance undergoes a specific change, which we'll discuss in the following sections. Remember, knowing how to calculate variance allows you to measure the reliability of data points within a dataset.

random variables

Random variables are essentially quantities whose values are subject to chance. They can be outcomes of a random phenomenon or experiment. Understanding random variables is crucial for probability and statistics as they help model real-world random processes. There are two types of random variables:

• Discrete Random Variables: Take on a countable number of distinct values, like the number of heads when flipping a coin.
• Continuous Random Variables: Can take an infinite number of possible values, like the exact weight of a randomly selected person.

The variance of a random variable, which describes the spread of its distribution, is key to understanding its behavior under different conditions.

linear transformation formula

Linear transformations are changes made to a random variable, often in the form of adding a constant or multiplying by a constant. The formula for the variance of a linear transformation of a random variable is:\[ Var(a + bX) = b^2 \times Var(X) \]where \( a \) and \( b \) are constants, and \( X \) is the random variable. It's important to note that only the scaling factor \( b \) (multiplication) affects the variance. The addition of a constant \( a \) does not impact it. In the provided exercise, knowing to identify \( a \) and \( b \) correctly and applying them to this formula correctly led us to find the transformed variance of 100. This understanding aids in many areas of statistics where transforming data can simplify complex problems.