5. Binomial Distribution & Discrete Random Variables
Discrete Random Variables
2:33 minutesProblem 4.1.41
Textbook Question
Linear Transformation of a Random Variable In Exercises 41 and 42, use this information about linear transformations. For a random variable x, a new random variable y can be created by applying a linear transformation , where a and b are constants. If the random variable x has mean and standard deviation , then the mean, variance, and standard deviation of y are given by the formulas
The mean annual salary of employees at an office is originally $46,000. Each employee receives an annual bonus of $600 and a 3% raise (based on salary). What is the new mean annual salary (including the bonus and raise)?
Verified step by step guidance1
Step 1: Understand the linear transformation formula. A new random variable y can be created using the formula y = a + b * x, where 'a' represents a constant addition (bonus) and 'b' represents a scaling factor (raise percentage).
Step 2: Identify the given values. The mean annual salary of employees (mean of x) is $46,000. The bonus (a) is $600, and the raise percentage (b) is 3%, which can be expressed as 0.03 in decimal form.
Step 3: Apply the formula for the mean of the transformed random variable y. The formula for the mean of y is given by: mean(y) = a + b * mean(x). Substitute the values of a, b, and mean(x) into this formula.
Step 4: Perform the substitution. Replace 'a' with 600, 'b' with 0.03, and 'mean(x)' with 46,000 in the formula mean(y) = a + b * mean(x).
Step 5: Simplify the expression to find the new mean annual salary. Combine the terms to calculate the new mean, which includes the bonus and raise.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Transformation
A linear transformation involves modifying a random variable using a linear equation of the form y = ax + b, where 'a' and 'b' are constants. This transformation affects the mean and standard deviation of the original variable. Specifically, the mean of the new variable y is calculated as E(y) = aE(x) + b, while the standard deviation is adjusted by the absolute value of 'a', given by SD(y) = |a|SD(x).
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Mean
The mean, or expected value, of a random variable is a measure of central tendency that represents the average outcome. It is calculated by summing all possible values of the variable, each weighted by its probability. In the context of salary, the mean provides a baseline for understanding the average earnings of employees before any adjustments like bonuses or raises are applied.
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Variance and Standard Deviation
Variance measures the spread of a set of values around the mean, indicating how much the values differ from the average. The standard deviation is the square root of the variance and provides a more interpretable measure of spread in the same units as the original data. In salary calculations, understanding variance and standard deviation helps assess the consistency of salaries among employees.
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