Answer: The resulting distribution will not be normal. In fact, it will be very heavily tailed.

## Explanation When we mix two normal distributions with the same mean but different variances, the resulting mixture distribution is **not normal**. Here's why: - **Heavy Tails**: The mixture distribution combines the properties of both component distributions. Since one distribution has variance 1 and the other has variance 3, the mixture will have heavier tails than a single normal distribution. This is because the higher variance component contributes more extreme values. - **Not Bimodal**: Option B is incorrect because bimodality typically occurs when the component distributions have different means. Here, both distributions have the same mean (0), so the mixture will not be bimodal. - **Not Skewed**: Option D is incorrect because the mixture of two symmetric distributions (both normal with mean 0) remains symmetric. There is no skewness introduced. - **Not Normal**: Option A is incorrect because the mixture of two normal distributions with different variances is not normal. While the mean remains 0, the variance is not simply the average (2), and more importantly, the shape is not normal. **Mathematical Insight**: The probability density function of the mixture is: \[ f(x) = 0.5 \cdot \frac{1}{\sqrt{2\pi}} e^{-x^2/2} + 0.5 \cdot \frac{1}{\sqrt{6\pi}} e^{-x^2/6} \] This distribution exhibits **leptokurtosis** (excess kurtosis > 0), meaning it has heavier tails than a normal distribution.

Author: LeetQuiz .