Math Problem Statement
Solution
Let's break down the question and analyze the statements based on the rules of variance:
-
The variance of :
When adding or subtracting a constant to a random variable, the variance remains unchanged. This is because the variance only measures the spread of the data and shifting it does not affect the spread. Therefore, the variance of is the same as the variance of , which is 2.
This statement is true. -
The variance of :
When multiplying a random variable by a constant , the variance is scaled by . So, the variance of is given by: This means the variance of is 8, not -4.
This statement is false.
Final Answer:
- The first statement is true.
- The second statement is false.
Would you like a further breakdown or more details?
Here are five related questions to deepen your understanding:
- What happens to the variance if you multiply a random variable by a constant?
- How does adding a constant affect the mean of a random variable?
- What is the standard deviation of if the standard deviation of is known?
- Can the variance of a random variable ever be negative?
- How do transformations like impact both the variance and mean of a random variable?
Tip: Variance is always non-negative, and it scales by the square of a constant when the random variable is multiplied by that constant.
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Math Problem Analysis
Mathematical Concepts
Statistics
Variance
Random Variables
Formulas
Var(X + c) = Var(X)
Var(aX) = a^2 * Var(X)
Theorems
Scaling property of variance
Addition of constants does not affect variance
Suitable Grade Level
Undergraduate level statistics
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