If a random variable X is defined such that E[(X - 1)²] = 10, and E[(X - 2)²] = 6. Find the mean of X.
Understand the Problem
The question is asking to find the mean of a random variable X given two expected values related to its deviations from specific numbers. We will apply concepts from statistics to derive the mean based on the provided equations.
Answer
The mean of \(X\) is $\mu = 3.5$.
Answer for screen readers
The mean of (X) is $\mu = 3.5$.
Steps to Solve
- Understand the equations for expected values
The given equations are: $$ E[(X - 1)^2] = 10 $$ $$ E[(X - 2)^2] = 6 $$
These express the expected values of the squared deviations of $X$ from 1 and 2.
- Expand the equations
We can expand the squared terms:
- For the first equation: $$ E[(X - 1)^2] = E[X^2 - 2X + 1] = E[X^2] - 2E[X] + 1 = 10 $$
- For the second equation: $$ E[(X - 2)^2] = E[X^2 - 4X + 4] = E[X^2] - 4E[X] + 4 = 6 $$
- Set up a system of equations
From the expansions, we can create two equations (denote $E[X]$ as $\mu$):
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$$ E[X^2] - 2\mu + 1 = 10 \quad \Rightarrow \quad E[X^2] = 2\mu + 9 \tag{1} $$
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$$ E[X^2] - 4\mu + 4 = 6 \quad \Rightarrow \quad E[X^2] = 4\mu + 2 \tag{2} $$
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Equate the equations for (E[X^2])
Set the right-hand sides of (1) and (2) equal to each other: $$ 2\mu + 9 = 4\mu + 2 $$
- Solve for (\mu)
Rearranging gives: $$ 9 - 2 = 4\mu - 2\mu $$ $$ 7 = 2\mu $$ Thus, $$ \mu = \frac{7}{2} = 3.5 $$
The mean of (X) is $\mu = 3.5$.
More Information
The mean of a random variable is a measure of the center of its distribution. In this case, manipulating the given expected values helped derive the mean effectively. This problem illustrates the use of properties of expected values and equations in statistics.
Tips
- Confusing the expected value notation with regular variable calculations.
- Not expanding the equations properly, which can lead to incorrect expressions.
- Forgetting to account for the constants when manipulating the equations.