For a chi-squared ($\chi^2$) distribution with degrees of freedom ($k > 1$), here’s the detailed behavior of the probability density function (PDF) and cumulative distribution function (CDF):

1. Probability Density Function (PDF):

   • Starts at $(x = 0, y = 0)$ since $f(x) \to 0$ as $x \to 0$ for $k > 1$.
   • Increases to a maximum: The PDF rises to its peak at a point dependent on $k$ and then starts to decrease.
   • Decreases to a horizontal asymptote: The PDF approaches $y = 0$ as $x \to \infty$, which is the horizontal asymptote.

2. Cumulative Distribution Function (CDF):

   • Starts at $(x = 0, y = 0)$ because no probability mass has been accumulated yet.
   • Increases: The CDF continuously rises as $x$ increases since it is the cumulative sum of probabilities.
   • Approaches a horizontal asymptote of $y = 1$: As $x \to \infty$, the total probability mass reaches 1.

Explanation:

• The shape of the PDF is skewed right for smaller $k$, with more pronounced asymmetry. As $k$ increases, the PDF becomes more symmetric, approaching a normal distribution due to the Central Limit Theorem.
• The CDF is always monotonic (non-decreasing), beginning at 0 and asymptotically reaching 1.

Would you like a graph or detailed derivation of any specific points for better visualization?

Related Questions:

1. How does the peak of the PDF change as degrees of freedom $k$ increase?
2. Why does the chi-squared distribution have a horizontal asymptote at $y = 0$ for the PDF?
3. What role does the gamma function play in defining the chi-squared distribution?
4. How does the chi-squared distribution relate to the normal distribution?
5. What are some practical applications of the chi-squared distribution in hypothesis testing?

Tip: The chi-squared distribution is commonly used in goodness-of-fit tests and tests for independence in contingency tables. Understanding its shape helps interpret these tests.