> 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/10-exponential.md).

# Exponential

* Exponential RV has domain $$\mathcal{D}=(0,\infty)$$ and pdf:

  $$
  f(x)=\frac{1}{\beta}e^{-x/\beta},\quad x\in \mathcal{D}.
  $$
* Only parameter is $$\beta \in \mathbb{R}\_+$$ (a.k.a. scale parameter).
* Reciprocal $$\eta=1/\beta$$ is known as intensity and thus pdf can also be written as $$f(x)=\eta e^{-\eta\cdot x}$$.
* Express fact that $$X$$ is exponential as $$X\sim \textrm{Exp}(\beta)$$.

## Exponential RV

* Pdf of exponential RV tells us that:
  * Probability of finding $$X$$ away from zero decays exponentially fast at rate $$\eta$$
  * There is *zero* probability of finding $$X$$ below zero (pdf is asymmetric).
* Cdf is $$F(x)=1-e^{-x/\beta}$$ and note $$1-F(x)=e^{-x/\beta}$$.
* Expected value and variance are $$\mathbb{E}\_X=\beta$$ and $$\mathbb{V}\_X=\beta^2$$.
* This RV is often used to model time-dependent phenomena (e.g., *failures*).
* For instance, $$X$$ can be used to model time that we have to wait until we observe first occurrence of an event (e.g., engine fails).
* In this context, we know average waiting time ($$\mathbb{E}\_X=\beta$$) but actual time is unknown.
* Parameter $$\eta=1/\beta$$ can be interpreted as event rate (how frequently they occur).

## Exponential

* Pdfs and cdfs for $$\textrm{Exp}(\beta)$$ for different values of $$\beta$$.

  <figure><img src="https://1753452229-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FakKE106umUCSWFtTnl31%2Fuploads%2Fgit-blob-adba2c426dbf8f323599b8871454e9e4e2fee01b%2Fch2_compareexponentials.png?alt=media" alt="" width="420"><figcaption></figcaption></figure>

{% hint style="info" %}
How to determine $$\beta$$ from cdf?
{% endhint %}

* Note that $$F(x)=1-e^{-1}=0.63$$ when $$x=\beta$$. This is known as characteristic time.

<details>

<summary>Python: reproduce this figure (ch2_compare_exponentials.ipynb)</summary>

Exponential RVs $$X\sim\mathrm{Exp}(\beta)$$ for $$\beta=1,2,4$$ (Victor Z and J E Umaña, UW-Madison, 2024). `scipy.stats.expon` uses `scale` $$=\beta$$ (the intensity is $$\eta=1/\beta$$). The dashed line marks $$F(\beta)=1-e^{-1}=0.63$$.

**Note:** This notebook reproduces the figure of the exponential RV in Section 2.2.4 of *Statistics for Chemical Engineers*, with realizations of Exp(4).

```python
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import expon

beta = [1, 2, 4]
lw = [0.5, 1, 1.5]
x = np.linspace(0, 4, 1000)

np.random.seed(0)
S = 1000
x_omega = np.random.exponential(scale=4, size=S)

plt.figure(figsize=(10, 8))
plt.subplot(2, 2, (1, 2))
plt.plot(x_omega, 'ko', markerfacecolor='w', markersize=4)
plt.xlabel(r'Outcome $\omega$', fontsize=14)
plt.ylabel(r'$x_\omega$', fontsize=14)
plt.axis([0, S, 0, 36])
plt.grid(True)
plt.subplot(2, 2, 3)
for k in range(len(beta)):
    plt.plot(x, expon.pdf(x, scale=beta[k]), 'k', linewidth=lw[k])
plt.grid(True)
plt.xlabel('$x$', fontsize=14)
plt.ylabel('$f(x)$', fontsize=14)
plt.axis([0, 4, 0, 1])
plt.subplot(2, 2, 4)
for k in range(len(beta)):
    plt.plot(x, expon.cdf(x, scale=beta[k]), 'k', linewidth=lw[k], label=rf'$\beta={beta[k]}$')
plt.axhline(1 - np.exp(-1), color='k', ls='--')
plt.grid(True)
plt.legend(loc='lower right', fontsize=10)
plt.xlabel('$x$', fontsize=14)
plt.ylabel('$F(x)$', fontsize=14)
plt.axis([0, 4, 0, 1])
plt.show()
```

<figure><img src="https://1753452229-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FakKE106umUCSWFtTnl31%2Fuploads%2Fgit-blob-59e549d7c07844d37a24e36ddfaa25b38780ad64%2Fch2_compare_exponentials-03-0.png?alt=media" alt="" width="520"><figcaption></figcaption></figure>

**Note:** The exponential RV models waiting times; its mean is the scale parameter beta.

Open in Colab: [`ch2_compare_exponentials.ipynb`](https://colab.research.google.com/github/vzavala/StatsBook/blob/master/Python/ch2_compare_exponentials.ipynb) · MATLAB live script: [`ch2_compare_exponentials.mlx`](https://github.com/vzavala/StatsBook/blob/master/Matlab-Live/ch2_compare_exponentials.mlx)

</details>

## Example: Residence Time in Mixing System

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

* Mixed system with volume $$V$$ and input/output flow $$F$$
* At time $$t=0$$ we inject tracer so that input concentration is $$C\_0$$.
* Tracer concentration in system at time $$t$$ is $$C(t)$$
  {% endhint %}

{% hint style="info" %}
For how long will a tracer particle reside in the system? What factors influence this time?
{% endhint %}

<figure><img src="https://1753452229-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FakKE106umUCSWFtTnl31%2Fuploads%2Fgit-blob-6b852105607631b7131d19f5e41d5aacdbb150fb%2Fparticle_diagram.png?alt=media" alt="" width="210"><figcaption></figcaption></figure>

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

* Material balance reveals that:

  $$
  f\_T(t)=\frac{C(t)}{C\_0}= \frac{1}{\tau}e^{-t/\tau};; \textrm{with};; \tau=V/Q
  $$
* Interpret $$T$$ as random time required for a particle to exit system (residence time)
* Pdf of $$T$$, $$f\_T(t)$$, is interpreted as fraction of injected particles exiting at time $$T=t$$
* Balance suggests that $$T\sim\textrm{Exp}(\tau)$$
  {% endhint %}

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

* Mean residence time is $$\mathbb{E}\[T]=\tau=V/Q$$ and variance is $$\mathbb{V}\[T]=(V/Q)^2$$.
* What is effect of $$V$$ and $$Q$$ on uncertainty?
* Fraction of particles that have exited up to time $$t$$ is $$\mathbb{P}(T\leq t)=F\_T(t)$$.
* $$F\_T(t)=\displaystyle\int\_{0}^tf\_T(t')dt'=(1-e^{-t/\tau})$$ and thus $$F\_T(0)=0$$ and $$F\_T(\infty)=1$$.
* Fraction of particles that remain in system at time $$t$$ is $$\mathbb{P}(T>t)=1-F\_T(t)=e^{-t/\tau}$$
* Function $$\mathbb{P}(T>t)$$ is known as the “survival function".
  {% endhint %}

#### Python implementation

{% hint style="success" %}
Notebook [`ch2_example_exponential.ipynb`](https://colab.research.google.com/github/vzavala/StatsBook/blob/master/Python/ch2_example_exponential.ipynb) — open it in Colab and run the cells alongside this section.

MATLAB live script: [`ch2_example_exponential.mlx`](https://github.com/vzavala/StatsBook/blob/master/Matlab-Live/ch2_example_exponential.mlx)
{% endhint %}

**Additional example: failure time of an adhesive** (Victor Z and J E Umaña, UW-Madison, 2024). The time to failure is $$X\sim\mathrm{Exp}(\beta)$$ with average $$\mathbb{E}\_X=\beta=36$$ months. The probability that the material survives 12 months is the survival function $$\mathbb{P}(X>12)=1-F(12)=e^{-12/\beta}$$. Python names: `beta`, realizations `x_omega`, pdf/cdf from `scipy.stats.expon(scale=beta)`.

**Note:** This notebook reproduces Example 2.16 (Section 2.2.4) of *Statistics for Chemical Engineers*: lifetime of an adhesive with an average of 36 months.

```python
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import expon

beta = 3 * 12            # months (scale; intensity eta = 1/beta)
X = expon(scale=beta)

np.random.seed(0)
S = 1000
x_omega = X.rvs(size=S)

x = np.linspace(0, 160, 400)
fig, axs = plt.subplots(3, 1, figsize=(8, 11))
axs[0].plot(x_omega, 'ko', markerfacecolor='w', markersize=4)
axs[0].set_xlabel(r'Outcome $\omega$', fontsize=14)
axs[0].set_ylabel(r'$x_\omega$ [months]', fontsize=14)
axs[0].axis([0, S, 0, 300])
axs[1].hist(x_omega, bins=np.arange(0, 161, 10), density=True, edgecolor='black', facecolor='none')
axs[1].plot(x, X.pdf(x), 'k', linewidth=1.5)
axs[1].set_xlabel('$x$ [months]', fontsize=14)
axs[1].set_ylabel('$f(x)$', fontsize=14)
axs[1].axis([0, 160, 0, 0.03])
axs[2].plot(x, X.cdf(x), 'k', linewidth=1.5)
axs[2].set_xlabel('$x$ [months]', fontsize=14)
axs[2].set_ylabel('$F(x)$', fontsize=14)
axs[2].axis([0, 160, 0, 1])
for ax in axs:
    ax.grid(True)
plt.tight_layout()
plt.show()

print(f"P(X > 12) = 1 - F(12) = {X.sf(12):.4f}  (e^(-12/beta) = {np.exp(-12/beta):.4f})")
```

<figure><img src="https://1753452229-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FakKE106umUCSWFtTnl31%2Fuploads%2Fgit-blob-716cf5e1312286c8545017d660929286d0317f1b%2Fch2_example_exponential-03-0.png?alt=media" alt="" width="520"><figcaption></figcaption></figure>

```
P(X > 12) = 1 - F(12) = 0.7165  (e^(-12/beta) = 0.7165)
```

**Note:** As in the book, to survive 12 months with probability 0.95 the average lifetime must be 233 months: about 6.5 times longer than now.