Estimation of Lower Bound in Poisson Distribution

Chi-Square (χ²) Explorer

Visualize degrees of freedom and critical regions.

Degrees of Freedom (k) Try k=2 vs k=10 to see shape change

---

Find Probability: Right Tail (P > x) [P-Value] Left Tail (P < x)

Chi-Square Value (x)

Probability (Area): 0.0500

Theory: The Chi-Square Distribution

Topic: Sampling Distributions | Difficulty: Intermediate

The Chi-Square distribution with $k$ degrees of freedom is the distribution of a sum of the squares of $k$ independent standard normal random variables. It is a fundamental distribution in statistical inference, used primarily in hypothesis testing (Goodness of Fit, Test of Independence).

Probability Density Function (PDF)

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

For $x > 0$ and $k > 0$. $\Gamma(\cdot)$ represents the Gamma function.

Moments

• Mean ($E[X]$): $k$
• Variance ($Var(X)$): $2k$
• Mode: $\max(0, k-2)$

Shape Parameters

• Skewness: $\sqrt{8/k}$ (Always Right Skewed)
• Kurtosis (Excess): $12/k$
• Range: $[0, +\infty)$

Key Properties

1. Additivity: If $X \sim \chi^2(k_1)$ and $Y \sim \chi^2(k_2)$ are independent, then $X+Y \sim \chi^2(k_1 + k_2)$.
2. Relation to Normal: As $k \to \infty$, the distribution converges to a Normal distribution (Central Limit Theorem).
3. Relation to F-Test: The ratio of two independent Chi-square variables divided by their degrees of freedom follows an F-distribution.

Python Implementation

Python (Scipy)

import scipy.stats as stats

df = 5   # Degrees of freedom
x = 11.07

# 1. Calculate P-Value (Right Tail Area)
# P(X > x) = 1 - CDF(x)
p_val = 1 - stats.chi2.cdf(x, df)
print(f"P-Value (Right Tail) = {p_val:.4f}")

# 2. Find Critical Value from Alpha
# Find x such that P(X > x) = 0.05
alpha = 0.05
crit_val = stats.chi2.ppf(1 - alpha, df)
print(f"Critical Value for alpha {alpha} = {crit_val:.4f}")