What is the value of $$ \beta_{1} $$ for $$ t $$-distribution?

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In the context of probability distributions, particularly when discussing **moments**, the parameter $$ \beta_{1} $$ typically refers to the **skewness** of the distribution. Skewness measures the asymmetry of the probability distribution of a real-valued random variable about its mean.

### **Skewness ($$ \beta_{1} $$) of the $$ t $$-Distribution**

The **$$ t $$-distribution** is a symmetric distribution centered around zero. This symmetry implies that the distribution has **no skewness**. Therefore, the skewness ($$ \beta_{1} $$) for the $$ t $$-distribution is:

$$
\beta_{1} = 0
$$

### **Key Points:**

1. **Symmetry:** The $$ t $$-distribution is symmetric about the mean (which is zero). This symmetry ensures that the left and right tails of the distribution are mirror images of each other.

2. **Skewness Interpretation:** 
   - **Positive Skewness ($$ \beta_{1} > 0 $$)**: The distribution has a longer or fatter tail on the right side.
   - **Negative Skewness ($$ \beta_{1} < 0 $$)**: The distribution has a longer or fatter tail on the left side.
   - **Zero Skewness ($$ \beta_{1} = 0 $$)**: The distribution is perfectly symmetric.

3. **Implications for $$ t $$-Distribution:**
   - Since the $$ t $$-distribution is symmetric, it doesn’t exhibit any skewness.
   - This property is consistent across all degrees of freedom for the $$ t $$-distribution.

### **Additional Notes:**

- **Kurtosis:** While $$ \beta_{1} $$ pertains to skewness, the **kurtosis** of the $$ t $$-distribution (which measures the "tailedness" of the distribution) is **greater than that of the normal distribution**, especially for lower degrees of freedom. This means the $$ t $$-distribution has heavier tails, allowing for more outliers.

- **Degrees of Freedom:** The properties of the $$ t $$-distribution, including its kurtosis, vary with different degrees of freedom. However, the skewness remains zero regardless of the degrees of freedom.

### **Conclusion**

For the $$ t $$-distribution, the skewness parameter $$ \beta_{1} $$ is **zero**, reflecting its inherent symmetry.
The skewness ($$ \beta_{1} $$) of the $$ t $$-distribution is 0, meaning it is perfectly symmetric.

What is the value of for -distribution?

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