> For the complete documentation index, see [llms.txt](https://zavalab.gitbook.io/statschemebook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://zavalab.gitbook.io/statschemebook/chapter-2-univariate-random-variables/12-chi-squared.md).

# Chi-Squared

Chi-Squared RV has pdf of the form:

$$
f(x)=\frac{1}{2^{r/2}\Gamma(r/2)}e^{-x/2}x^{r/2-1},; x\in \mathcal{D}.
$$

with domain $$\mathcal{D}=\[0,\infty)$$.

* Pdf has only one parameter $$r\in \mathbb{Z}\_+$$ (a.k.a. degrees of freedom).
* We express fact that $$X$$ is a chi-squared RV as $$X\sim \chi^2(r)$$
* This is a Gamma RV with $$\beta=2$$ and $$\alpha=r/2$$ (for a positive integer $$r$$).
* Expected value and variance can be derived from those of the Gamma RV.
* Crucial property of a chi-squared RV is that it is related to standard normal RV. In particular, one can show that:

  $$
  \sum\_{i=1}^rX\_i^2\sim \chi^2(r)
  $$

  if $$X\_i\sim \mathcal{N}(0,1)$$ and $$X\_i$$ are independent. This property will be useful later on.
* From relationship with $$\mathcal{N}(0,1)$$ we can show that $$\mathbb{Q}\_Y(1-\alpha)$$ of $$Y\sim\chi^2(1)$$ is equal to $$\mathbb{Q}\_Z(1-\alpha/2)^2$$ of $$Z\sim \mathcal{N}(0,1)$$.
* This is because $$\mathbb{P}(-z\leq Z\leq z)=\mathbb{P}(Y\leq z)$$ with $$Y=Z^2$$.

<details>

<summary>Python: reproduce this figure (ch2_compare_chisq.ipynb)</summary>

Chi-squared RVs $$X\sim\chi^2(r)$$ as Gamma RVs with $$\beta=2$$, $$\alpha=r/2$$ (Victor Z and J E Umaña, UW-Madison, 2024).

**Note:** This notebook supports Section 2.2.6 of *Statistics for Chemical Engineers*: it compares the quantile functions of chi2(1) and N(0,1).

```python
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import gamma, chi2, norm

x = np.linspace(0, 4, 1000)
plt.figure(figsize=(10, 4))
for sub, fun, label in ((1, gamma.pdf, '$f(x)$'), (2, gamma.cdf, '$F(x)$')):
    plt.subplot(1, 2, sub)
    for r in (1, 2, 3, 4):
        plt.plot(x, fun(x, r/2, scale=2), linewidth=1.5, label=f'$r={r}$')   # = chi2(r)
    plt.grid(True)
    plt.xlabel('$x$', fontsize=12)
    plt.ylabel(label, fontsize=12)
    plt.axis([0, 4, 0, 1])
plt.legend(loc='lower right', fontsize=9)
plt.tight_layout()
plt.show()
```

<figure><img src="https://1753452229-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FakKE106umUCSWFtTnl31%2Fuploads%2Fgit-blob-9cbfb169e095d0a762c93e9b6b7139be205a0a96%2Fch2_compare_chisq-01-0.png?alt=media" alt="" width="520"><figcaption></figcaption></figure>

$$\mathbb{Q}\_Y(1-\alpha)$$ for $$Y\sim\chi^2(1)$$ equals $$\mathbb{Q}\_Z(1-\alpha/2)^2$$ for $$Z\sim\mathcal{N}(0,1)$$.

```python
alpha = np.linspace(0.01, 1, 100)
plt.figure(figsize=(10, 4))
plt.subplot(1, 2, 1)
plt.plot(alpha, chi2.ppf(1 - alpha, df=1), 'b-', linewidth=1.5)
plt.ylabel(r'$\mathbb{Q}_Y(1-\alpha)$', fontsize=12)
plt.subplot(1, 2, 2)
plt.plot(alpha, norm.ppf(1 - alpha/2)**2, 'b-', linewidth=1.5)
plt.ylabel(r'$\mathbb{Q}_Z(1-\alpha/2)^2$', fontsize=12)
for k in (1, 2):
    plt.subplot(1, 2, k)
    plt.grid(True)
    plt.xlabel(r'$\alpha$', fontsize=12)
    plt.axis([0, 1, 0, 8])
plt.tight_layout()
plt.show()
```

<figure><img src="https://1753452229-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FakKE106umUCSWFtTnl31%2Fuploads%2Fgit-blob-2f1215033dcc86a106d5209320b46b0290df5791%2Fch2_compare_chisq-03-0.png?alt=media" alt="" width="520"><figcaption></figcaption></figure>

**Note:** The square of a N(0,1) RV is a chi2(1) RV, so the quantiles of chi2(1) give the critical values of the Gaussian.

Open in Colab: [`ch2_compare_chisq.ipynb`](https://colab.research.google.com/github/vzavala/StatsBook/blob/master/Python/ch2_compare_chisq.ipynb) · MATLAB live script: [`ch2_compare_chisq.mlx`](https://github.com/vzavala/StatsBook/blob/master/Matlab-Live/ch2_compare_chisq.mlx)

</details>

## Example: Computing Confidence Intervals

{% hint style="danger" icon="flask" %}

* Consider output flow $$F\_3\sim \mathcal{N}(30,1)$$.
* Compute 90% confidence interval for $$F\_3$$ using quantile of Chi-Squared RV.
* Confidence interval is $$\mu\pm z\_{\alpha/2}\sigma$$ with $$z\_{\alpha/2}=\mathbb{Q}\_Z(1-\alpha/2)$$ and $$Z\sim \mathcal{N}(0,1)$$.
* Instead of using $$Z\sim \mathcal{N}(0,1)$$, here we compute $$z\_{\alpha/2}$$ using $$Y\sim \chi^2(1)$$ as:

  $$
  \mathbb{Q}*{Z}(1-\frac{\alpha}{2})=z*{\alpha/2}=\sqrt{\mathbb{Q}\_Y(1-\alpha)}
  $$
* We use numerical package to compute $$\mathbb{Q}*Y(1-0.1)=2.71$$ and thus $$z*{\alpha/2}=\sqrt{2.71}=1.64$$.
* Confidence interval is $$\[30\pm 1\cdot 1.64]=\[28.36,31.64]$$ gpm. This is the same confidence interval computed previously using the Gaussian RV model.
  {% endhint %}

#### Python implementation: 90% interval for the mixer output

{% hint style="success" %}
Notebook [`ch2_example_chisquare.ipynb`](https://colab.research.google.com/github/vzavala/StatsBook/blob/master/Python/ch2_example_chisquare.ipynb) — open it in Colab and run the cells alongside this section.

MATLAB live script: [`ch2_example_chisquare.mlx`](https://github.com/vzavala/StatsBook/blob/master/Matlab-Live/ch2_example_chisquare.mlx)
{% endhint %}

Confidence intervals and regions from chi-squared quantiles (Victor Z and J E Umaña, UW-Madison, 2024).

| Slides                                           | Python        | Meaning                          |
| ------------------------------------------------ | ------------- | -------------------------------- |
| $$F\_3\sim\mathcal{N}(\mu,\sigma^2)$$            | `mu`, `sigma` | output flow of the mixer         |
| $$\alpha$$                                       | `alpha`       | probability outside the interval |
| $$Y\sim\chi^2(1)$$, $$\mathbb{Q}\_Y(1-\alpha)$$  | `Q_Y`         | chi-squared quantile             |
| $$z\_{\alpha/2}=\sqrt{\mathbb{Q}\_Y(1-\alpha)}$$ | `z_alpha_2`   | critical value                   |

**Note:** This notebook reproduces Example 2.19 (Section 2.2.6) of *Statistics for Chemical Engineers*: confidence regions for two independent Gaussian RVs.

```python
import sys
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import chi2, norm

mu, sigma = 30, 1
alpha = 0.1
Q_Y = chi2.ppf(1 - alpha, df=1)            # Q_Y(1 - alpha), Y ~ chi2(1)
z_alpha_2 = np.sqrt(Q_Y)                   # equals Q_Z(1 - alpha/2)
print(f"Q_Y(0.9) = {Q_Y:.2f},  z_alpha/2 = {z_alpha_2:.2f}  (normal: {norm.ppf(1 - alpha/2):.2f})")
print(f"90% interval: [{mu - z_alpha_2*sigma:.2f}, {mu + z_alpha_2*sigma:.2f}] gpm")
```

```
Q_Y(0.9) = 2.71,  z_alpha/2 = 1.64  (normal: 1.64)
90% interval: [28.36, 31.64] gpm
```

***

#### Python implementation: Going further: a 2D confidence ellipse from chi2(2)

For a 2D standard normal $$Z=(Z\_1,Z\_2)$$, $$Z^T\Sigma^{-1}Z\sim\chi^2(2)$$, so the region $${z: z^T\Sigma^{-1}z\leq \mathbb{Q}\_{\chi^2(2)}(1-\alpha)}$$ holds $$Z$$ with probability $$1-\alpha$$. (Multivariate Gaussians are covered in Chapter 3.) The `ellipse` helper is in `Support/ellipse.py`.

```python
sys.path.append(path + 'Support')
from ellipse import ellipse

mu_vec = np.array([0, 0])
Sigma = np.array([[1, 0], [0, 1]])
level = chi2.ppf(1 - alpha, 2)                       # chi2(2) quantile
xe, ye, _, _, _ = ellipse(np.linalg.inv(Sigma), level, 100, mu_vec)
r1 = np.sqrt(chi2.ppf(1 - alpha, 1))                 # 1D interval half-width

plt.figure(figsize=(6, 6))
plt.plot(xe, ye, 'k', linewidth=1.5, label=r'$\chi^2(2)$ region')
plt.plot([-r1, r1], [0, 0], 'k--', linewidth=1.0, label=r'$\chi^2(1)$ intervals')
plt.plot([0, 0], [-r1, r1], 'k--', linewidth=1.0)
plt.xlabel('$z_1$', fontsize=14)
plt.ylabel('$z_2$', fontsize=14)
plt.axis([-2.5, 2.5, -2.5, 2.5])
plt.legend(loc='upper right')
plt.grid(True)
plt.show()

np.random.seed(1)
S = 1000
z_omega = np.random.multivariate_normal(mu_vec, Sigma, S)
inside = np.einsum('ij,jk,ik->i', z_omega - mu_vec, np.linalg.inv(Sigma), z_omega - mu_vec) <= level
print(f"{inside.sum()} of {S} samples fall in the ellipse (expected about {S*(1-alpha):.0f})")
```

<figure><img src="https://1753452229-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FakKE106umUCSWFtTnl31%2Fuploads%2Fgit-blob-7b93d1930c374e235c71507657679c5d2c105c81%2Fch2_example_chisquare-07-0.png?alt=media" alt="" width="520"><figcaption></figcaption></figure>

```
897 of 1000 samples fall in the ellipse (expected about 900)
```

**Note:** As in the book, the 90% region for (Z1, Z2) is a circle obtained from the chi-squared RV; the individual 90% regions of Z1 and Z2 are segments along the axes of that circle.