The graph you've uploaded shows a bell-shaped normal distribution curve.

To match the curve to its mean $\mu$ and standard deviation $\sigma$, let's analyze it based on the properties of a normal distribution:

1. Mean $\mu$: This determines the center of the curve. From the graph, the center appears to be at $x = 3$, so the mean $\mu = 3$.

2. Standard Deviation $\sigma$: This determines the spread or width of the curve. A smaller standard deviation means a narrower curve, while a larger standard deviation means a wider curve.

Comparing the choices:

• For $\mu = 3, \sigma = 3$: This will be a narrower curve because the smaller standard deviation indicates less spread.
• For $\mu = 3, \sigma = 6$: This will be a wider curve because the larger standard deviation indicates more spread.

In the uploaded graph, the curve is relatively narrow, which suggests that the standard deviation is smaller. Therefore, the graph likely corresponds to $\mu = 3$ and $\sigma = 3$.

Would you like any more details on normal distributions or further clarifications?

Here are 5 related questions to expand your understanding:

1. How does the standard deviation affect the shape of a normal distribution curve?
2. What is the empirical rule (68-95-99.7) in the context of a normal distribution?
3. How do you calculate the probability of an event given a normal distribution's mean and standard deviation?
4. What are the properties of a standard normal distribution (with $\mu = 0$ and $\sigma = 1$)?
5. How do transformations like shifting and scaling affect a normal distribution?

Tip: The area under the curve of a normal distribution always equals 1, which represents the total probability.