The
Incorrect answer:
Variable
of a probability distribution is given by the formula
. \(E[(x-\mu _{x})^{2}\cdot P(x)]\)

It seems like you're trying to describe the **variance** of a probability distribution, but there are some details missing in the formula you've written. Let me clarify it.

The **variance** \( \sigma^2 \) of a random variable \( X \) in a probability distribution is given by:

\[
\text{Var}(X) = E[(X - \mu_X)^2] = \sum (x - \mu_X)^2 \cdot P(x)
\]

Where:
- \( X \) is the random variable.
- \( \mu_X \) is the expected value (or mean) of the distribution, \( \mu_X = E[X] \).
- \( P(x) \) is the probability of each value \( x \).
- \( E[(X - \mu_X)^2] \) represents the expected value of the squared deviation from the mean.

In the corrected form, this formula expresses the variance of a discrete probability distribution, measuring how much the values of \( X \) deviate from the mean \( \mu_X \), weighted by their probabilities.

Would you like more details or have any other questions?

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Here are five related questions to explore further:
1. How do you calculate the mean \( \mu_X \) of a probability distribution?
2. What is the relationship between variance and standard deviation?
3. How does the formula for variance differ in continuous vs. discrete distributions?
4. Can you compute the variance for a specific probability distribution example?
5. What is the interpretation of variance in terms of data spread?

**Tip:** Variance is always non-negative because it involves squaring deviations, making the smallest possible value zero (when all outcomes are the same).

The variable of a probability distribution is given by the formula E[(x - μ_x)^2 · P(x)].

Explore the formula for calculating the variance of a probability distribution, including key concepts such as expected value, deviation from the mean, and how variance measures data spread. This detailed guide is suitable for college students or advanced high school learners.

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