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We are given the first three moments about the origin for a distribution:

\[
\mu'_1 = 0.5, \quad \mu'_2 = 0.3, \quad \mu'_3 = 0.2
\]

We need to determine which statement is true based on these moments.

### Key Points:
- **Mean (First Moment):** \(\mu'_1 = 0.5\), which is the mean.
- **Variance (Second Moment):** The second moment about the mean (which gives the variance) is \(\mu'_2 - (\mu'_1)^2 = 0.3 - (0.5)^2 = 0.3 - 0.25 = 0.05\). Hence, the variance is 0.05.
- **Skewness (Third Moment):** Skewness can be inferred from the third moment. Skewness is a measure of asymmetry. For a perfectly symmetric distribution, the third moment would be zero.

Given that \(\mu'_3 = 0.2\), we have a positive third moment, indicating that the distribution is **positively skewed**.

### Evaluating the options:
- **(a) Mean is less than variance:** The mean is \(0.5\) and the variance is \(0.05\). Clearly, the mean is greater than the variance, so this statement is false.
- **(b) The distribution is symmetric:** The skewness is non-zero (positive skewness), so the distribution is not symmetric. This is false.
- **(c) The distribution is positively skewed:** The third moment is positive (\(\mu'_3 = 0.2\)), indicating positive skewness. This is true.
- **(d) The distribution is negatively skewed:** Since the third moment is positive, the distribution cannot be negatively skewed. This is false.

### Correct Answer:
**(c) The distribution is positively skewed.**

Let me know if you need further details or have any questions!

Here are some related questions to consider:
1. What is the relationship between the moments and the shape of a distribution?
2. How does skewness affect the interpretation of data in a real-world scenario?
3. How are moments about the origin different from moments about the mean?
4. How can we compute higher-order moments for a given dataset?
5. Can a distribution have both positive and negative skewness at different intervals?

**Tip:** The third central moment (\(\mu_3\)) measures skewness, while the second central moment (\(\mu_2\)) measures variance, giving you insights into both spread and asymmetry.

The first three moments about origin of a distribution are μ'_1 = 0.5, μ'_2 = 0.3 and μ'_3 = 0.2. Then, which of the following statement is true?

Learn how to analyze the moments of a distribution. Given the moments μ'_1 = 0.5, μ'_2 = 0.3, and μ'_3 = 0.2, this problem focuses on calculating variance and determining skewness, concluding that the distribution is positively skewed. Suitable for undergraduate students studying statistics.

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