Chi-Square Distribution Density Evaluator

The chi-square distribution (denoted χ²) is one of the most important continuous probability distributions in statistical inference. It plays a pivotal role in hypothesis testing, confidence interval construction, and goodness-of-fit analysis. This distribution arises naturally as the sum of squares of independent standard normal random variables.

Given a sample observation point x and the degrees of freedom parameter k, your task is to compute the probability density function (PDF) value at that point.

Mathematical Foundation

The probability density function of the chi-square distribution is defined as:

$$f(x; k) = \frac{1}{2^{k/2} \cdot \Gamma(k/2)} \cdot x^{(k/2) - 1} \cdot e^{-x/2}$$

Where:

• x is the sample value (must be positive: x > 0)
• k is the degrees of freedom (a positive integer)
• Γ (Gamma) is the Gamma function, which extends the factorial to real and complex numbers
• e is Euler's number (approximately 2.71828)

Understanding the Gamma Function

The Gamma function is defined as: $$\Gamma(n) = (n-1)!$$ for positive integers

For half-integers: $$\Gamma(1/2) = \sqrt{\pi}$$

More generally: $$\Gamma(n + 1/2) = \frac{(2n-1)!!}{2^n} \sqrt{\pi}$$

Where !! denotes the double factorial.

Key Properties

1. Shape Behavior: The shape of the chi-square distribution depends entirely on the degrees of freedom parameter k
2. Mean: The expected value equals k
3. Variance: The variance equals 2k
4. Skewness: The distribution is right-skewed, with skewness decreasing as k increases
5. Limiting Behavior: As k → ∞, the chi-square distribution approaches a normal distribution

Your Task

Implement a function that calculates the probability density of a chi-square distribution at a given observation point x with k degrees of freedom. Return the result rounded to 3 decimal places.