.. _6-3-cdfs:

.. raw:: html

   <div class="video-placeholder" role="group" aria-labelledby="video-ch6-3">
     <iframe
       id="video-ch6-3"
       title="STAT 350 – Chapter 6.3 Cumulative Distribution Function Video"
       src="https://www.youtube.com/embed/cq_a1PFV0wQ?list=PLHKEwTHXfbagA3ybKLAcEJriGT-6k89c6"
       allowfullscreen>
     </iframe>
   </div>


Cumulative Distribution Functions
==============================================

Computing probabilities for continuous random variables requires integrating the PDF over intervals, 
which can become tedious when we need to calculate many different probabilities. Just as calculus 
provides us with antiderivatives to avoid repeated integration, probability theory offers us the 
cumulative distribution function (CDF) as a tool that eliminates the need for repeated 
integrations.

.. admonition:: Road Map 🧭
   :class: important

   • Define the **cumulative distribution function (CDF)**.
   • Master the **relationship between PDF and CDF** through the fundamental theorem of calculus.
   • Learn essential **properties of CDFs** that make them valid probability functions.
   • Practice **computing probabilities** using CDFs instead of direct integration.
   • Apply CDFs to find **percentiles and quantiles** for continuous distributions.

The Universal Language of Cumulative Probability
-------------------------------------------------

The **cumulative distribution function** (CDF) provides a unified approach to probability that 
works seamlessly for both discrete and continuous random variables. Instead of dealing 
with probability mass functions and probability density functions separately, the CDF 
gives us one consistent framework.

Definition
~~~~~~~~~~~~
For any random variable :math:`X` (discrete or continuous), the cumulative distribution 
function :math:`F_X(x)` is defined as:

.. math::
   F_X(x) = P(X \leq x)

The definition expresses a simple idea: :math:`F_X(x)` gives the probability 
that the random variable :math:`X` falls at or below any 
given value :math:`x`.

.. flat-table::
   :header-rows: 2
   :align: center
   :width: 100%

   * - :cspan:`1` Implementation of CDF to Different Variable Types
   
   * - Discrete 
     - Continuous

   * - .. math:: F_X(x) = P(X \leq x) = \sum_{t=-\infty}^{x} p_X(t)
     - .. math:: F_X(x) = P(X \leq x) = \int_{-\infty}^{x} f_X(t) \, dt

In both cases, the argument of the PMF/PDF is replaced with a dummy variable :math:`t`.
This is to avoid any confusion with teh true argument of :math:`F_X`.

In the continuous case, :math:`F_X(x)` can be used for :math:`P(X\leq x)` or
:math:`P(X<x)`

Requirements for a valid CDF
------------------------------

Cumulative distribution functions must satisfy several mathematical properties that 
reflect their probabilistic interpretation.

1. Monotonicity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

As we move from a smaller to a larger value, we can only **accumulate** probability. 
Thefere, :math:`F_X` is a non-decreasing function. That is, for any real numbers :math:`a < b`,

.. math::

   F_X(a) \leq F_X(b)

2. Limiting Behavior
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The accumulated probability must be close to 0 at the far left of the x-axis 
(no probability accumulates infinitely far to the left), 
and close to 1 at the far right (all probability accumulates when we include everything):

.. math::

   \lim_{x \to -\infty} F_X(x) = 0 \quad \text{and} \quad \lim_{x \to +\infty} F_X(x) = 1

3: Right Continuity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For any point :math:`c`,

.. math::

   \lim_{x \to c^+} F_X(x) = F_X(c).

This technical property ensures that the CDF behaves consistently at boundary points, 
particularly important when dealing with mixed discrete-continuous distributions or 
piecewise functions. This property is satisfied automatically for 
continuous distributions.

The PDF-CDF Connection
--------------------------------------------------

For continuous random variables, the relationship between the PDF and CDF mirrors 
the fundamental theorem of calculus.

From PDF to CDF: Integration
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

By the definition,

.. math::

   F_X(x) = \int_{-\infty}^{x} f_X(t) \, dt

:math:`F_X(x)` the **accummulation** of all the probability from the left tail of 
the distribution up to the point :math:`x`.

From CDF to PDF: Differentiation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Given a CDF :math:`F_X(x)`, we can recover the PDF by differentiation:

.. math::

   f_X(x) = \frac{d}{dx} F_X(x) = F_X'(x)

wherever the derivative exists. This relationship explains why the PDF measures 
the "rate of accumulation" of probability at each point.

Visualizing the Relationship
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The connection becomes clear when we graph the two functions together. 

.. _pdf-cdf-relationship:
.. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter6/pdf-cdf-comparison.png
   :alt: Relationship between PDF and CDF
   :align: center
   :figwidth: 70%

   Relationship between PDF and CDF

* If one function is piecewise-defined, the other is as well. They
  share the same set of boundaries.

* A probability of the form :math:`P(X\leq a)` corresponds to **the area to the left of**
  :math:`a` **under a PDF**, and to **the height of the curve** on a CDF.

* A PDF can take any shape, as long as it stays above the horizontal axis integrates to 1.
  A CDF is more constrained in shape; it approaches 0 on the left, 1 on the right, and is
  never decreasing in between.

* In any region where the PDF is :math:`0`, the CDF is flat, but not necessarily 0.
  It retains any area accummulated in previous regions, but no **additional** area is added. 




Building a Piecewise CDF
------------------------------

Constructing a CDF based on a piecewise-defined PDF requires careful attention.
The most crucial aspect of working with piecewise CDFs is remembering that they 
are **cumulative**—they must account for **all** probability accumulated up to each 
point, **not just in the current region**.

**Useful tips for constructing a piecewise CDF:**

* **Make a rough sketch** similar to :numref:`pdf-cdf-relationship`. Visually
  identify the regions where the CDF is approaching 0, incresing, flat, or
  approaching 1. At the end, check whether the computational result matches the sketch.

* **Stick to the definition** at first. By definition, 
   
  .. math:: F_X(x) = \int_{-\infty}^x f_X(t) \, dt

  is an integral from **negative infinity** up to the input value, :math:`x`.
  Starting directly from this definition-without skipping steps-will 
  help you become familiar with the subtle patterns involved in constructing piecewise CDFs.

.. admonition:: Example💡: Constructing the CDF for a piecewise PDF
   :class: note 

   Construct the CDF of a random variable :math:`X` which has the
   following PDF. Use :numref:`pdf-cdf-relationship` as reference.

   .. math::

      f_X(x) = 
      \begin{cases}
      \frac{3x}{2} & \text{for } 0 \leq x \leq 1 \\
      0 & \text{for } 1 < x < 5 \\
      \frac{1}{4} & \text{for } 5 \leq x \leq 6 \\
      0 & \text{elsewhere}
      \end{cases}

   Work on each region given by the PDF separately:

   **Region 1** (:math:`x < 0`): No probability accumulated yet

      .. math:: 
         F_X(x) = \int_{-\infty}^x f_X(t) \, dt = \int_{-\infty}^x 0 \, dt = 0

   **Region 2** (:math:`0 \leq x \leq 1`): Accumulating triangular area
      
      .. math:: 
         F_X(x) = \int_{-\infty}^x f_X(t) \, dt 
         = \underbrace{\int_{-\infty}^0 f_X(t)dt}_{=F_X(0) = 0} + \int_0^x \frac{3t}{2} \, dt = \frac{3x^2}{4}

   **Region 3** (:math:`1 < x < 5`): No new probability, but keep previous accumulation

      .. math:: 
         F_X(x) &= \int_{-\infty}^x f_X(t) \, dt \\
         &= \underbrace{\int_{-\infty}^0 f_X(t)dt + \int_0^1 \frac{3t}{2} \, dt}_{=F_X(1) 
         = \frac{3}{4}} + \underbrace{\int_1^x f_X(t) \, dt}_{=0} = \frac{3}{4}
      
      The area is constant at the triangle's total area.

   **Region 4** (:math:`5 \leq x \leq 6`): Add rectangular area to previous accumulation

      .. math:: 
         F_X(x) &= \int_{-\infty}^x f_X(t) \, dt \\
         &= \underbrace{\int_{-\infty}^0 f_X(t)dt + \int_0^1 \frac{3t}{2} \, dt 
         + \int_1^5 f_X(t) \, dt}_{=F_X(5)=\frac{3}{4}} + \int_5^x f_X(t) \, dt\\
         &=\frac{3}{4} + \int_5^x \frac{1}{4} \, dt = \frac{3}{4} + \frac{x-5}{4} = \frac{x-2}{4}

   **Region 5** (:math:`x > 6`): All probability has been accumulated 

      .. math:: 
         F_X(x) &= \int_{-\infty}^x f_X(t) \, dt \\
         &= F_X(6) + \int_{6}^{\infty} f_X(t) \, dt = 1 + \int_6^{\infty} 0 \, dt = 1.

   Putting together,

   .. math::

      F_X(x) =
      \begin{cases}
      0 & \text{for } x < 0 \\
      \frac{3x^2}{4} & \text{for } 0 \leq x \leq 1 \\
      \frac{3}{4} & \text{for } 1 < x < 5 \\
      \frac{x-2}{4} & \text{for } 5 \leq x \leq 6 \\
      1 & \text{for } x > 6
      \end{cases}

   * The video below goes through the series of examples covered in this section.

   .. raw:: html

      <div class="video-placeholder" role="group" aria-labelledby="video-ch6-3-2">
         <iframe 
         id="video-ch6-3-1"
         title="STAT 350 – Chapter 6.3.1 Example 1 Video"
         src="https://www.youtube.com/embed/hbpqL-h0830?si=YS-QR4AgqDrS_2EZ" 
         allowfullscreen></iframe>
      </div>

.. admonition:: 🛑 Know how to avoid the common mistake before simplifying steps
   :class: danger

   In all regions beyond the first in the previous example, 
   the CDF :math:`f_X(x)` is computed as the sum of area accumulated in the earlier regions **plus** 
   the integral over the current region. 

   Forgetting to factor in the area from previous regions is a very common mistake made by students.
   While you may eventually skip some redundant steps as you gain familiarity, 
   your top priority should be to avoid this error.

Computing Probabilities with CDFs
------------------------------------

Once we have the CDF, calculating probabilities becomes remarkably straightforward. 
We can handle any probability question without additional integration.

.. admonition:: Recall that equality signs do not matter for continuous RVs
   :class: important 

   All the rules below will apply if :math:`<` and :math:`\leq` are interchanged
   as well as if :math:`>` and :math:`\geq` are switched
   because :math:`P(X = a) = 0` for any single point :math:`a`.

Basic CDF Evaluation
~~~~~~~~~~~~~~~~~~~~~~~~

For :math:`P(X \leq a)`,

.. math::

   P(X \leq a) = F_X(a)

This is just the definition of the CDF—plug in the value and read the result.

"Greater Than" Probabilities
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For :math:`P(X > a)`, we use the complement rule:

.. math::

   P(X > a) = 1 - P(X \leq a) = 1 - F_X(a)

The probability of exceeding :math:`a` equals 
one minus the probability of not exceeding :math:`a`.

Interval Probabilities
~~~~~~~~~~~~~~~~~~~~~~~~

For :math:`P(a < X \leq b)`, we take the 
accumulated probability up to :math:`b` and **subtract** the accumulated probability 
up to :math:`a`, leaving just the probability in the interval.

.. math::

   P(a < X \leq b) = F_X(b) - F_X(a)

.. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter6/interval-prob-cdf.png
   :alt: Computing interval probabilities with CDFs
   :align: center
   :figwidth: 80%

   *P(a < X ≤ b) equals the difference between CDF values*

.. admonition:: Example💡: Computing Probabilities Using a CDF
   :class: note 

   Use the PDF and CDF found in the previous example to answer the questions below.

   .. math::

      f_X(x) = \begin{cases}
      \frac{3x}{2} & \text{for } 0 \leq x \leq 1 \\
      \frac{1}{4} & \text{for } 5 \leq x \leq 6 \\
      0 & \text{elsewhere}
      \end{cases}

   .. math::

      F_X(x) = \begin{cases}
      0 & \text{for } x < 0 \\
      \frac{3x^2}{4} & \text{for } 0 \leq x < 1 \\
      \frac{3}{4} & \text{for } 1 \leq x < 5 \\
      \frac{x-2}{4} & \text{for } 5 \leq x \leq 6 \\
      1 & \text{for } x > 6
      \end{cases}

   1. Find :math:`P(X \leq 0.5)`.
      
      :math:`x=0.5` belongs to Region 2 of the CDF, so :math:`F_X(0.5) = \frac{3(0.5)^2}{4} = \frac{3}{16}`.

   2. Find :math:`P(X \geq 5.5)`.

      :math:`P(X \geq 5.5) = 1 - P(X < 5.5) = 1 - F_X(5.5)`. :math:`x=5.5` is in Region 4.
      Therefore, :math:`P(X > 5.5) = 1 - \frac{5.5-2}{4} = 1 - \frac{7}{8} = \frac{1}{8}`

   3. Find :math:`P(0.2 < X \leq 5.8)`.

      .. math:: 
         P(0.2 < X \leq 5.8) &= F_X(5.8) - F_X(0.2)
         = \frac{5.8-2}{4} -\frac{3(0.2)^2}{4}\\ &= 0.95 - 0.03 = 0.92.

Finding Percentiles using CDF
-----------------------------------

Definition
~~~~~~~~~~~~~

Take a value :math:`p \in [0,1]`. The :math:`p \times 100` th percentile of a 
random variable :math:`X`, denoted :math:`x_p`, is the value satisfying:

.. math::

   F_X(x_p) = P(X \leq x_p) = p.

For example,

* The **50th percentile** of a random variable :math:`X` is written as :math:`x_{0.5}`
  and satisfies the conditon:
  
  .. math:: F_X(x_{0.5}) = P(X \leq x_{0.5})= 0.5.
  
  :math:`x_{0.5}` represents the vertical cutoff on the PDF of :math:`X`
  such that the area to its left is exactly :math:`0.5`.
  This special percentile is also called the **median** (:math:`\tilde{\mu}`) of :math:`X`.

  .. figure:: https://yjjpfnblgtrogqvcjaon.supabase.co/storage/v1/object/public/stat-350-assets/images/chapter6/50th-percentile.png
   :alt: 50th percentile
   :align: center
   :figwidth: 60%

* The **25th percentile** :math:`x_{0.25}` satisfies :math:`F_X(x_{0.25}) = 0.25`.  
  It splits the PDF into a left region with area 0.25 and
  right region with area 0.75.

* The **75th percentile** :math:`x_{0.75}` satisfies :math:`F_X(x_{0.75}) = 0.75`.  
  It splits the PDF into a left region with area 0.75 and
  right region with area 0.25.

* The *true* IQR of :math:`X` is the difference of :math:`x_{0.75} - x_{0.25}`.

.. admonition:: Connection to *sample* percentiles
   :class: important 
   
   Recall that we've already discussed the concept of *sample* percentiles for
   a *dataset* in Chapter 3. The percentiles for a *random variable* are considered 
   their *population* equivalent in the framework of data analysis.  
   
   If we generate data points :math:`x_1, x_2,\cdots, x_n` from
   a random variable :math:`X`, we would compute the *sample* percentiles using 
   :math:`x_1, x_2,\cdots, x_n`, and the *true* percentiles using the CDF
   of :math:`X`. The two sets of objects are closely related, but different.

Computing Percentiles of a Continuous Random Variable
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Percentiles for a continuous random variables are found by **replacing the 
general definition** :math:`F_X(x_p) = p` **with specific expressions, than solving for** :math:`x_p`.
See the example below for further detail.

.. admonition:: Example💡: Finding a Percentile
   :class: note 

   Find the 85th percentile of a random variable :math:`X` which has the
   following CDF:

   .. math::

      F_X(x) = \begin{cases}
      0 & \text{for } x < 0 \\
      \frac{3x^2}{4} & \text{for } 0 \leq x < 1 \\
      \frac{3}{4} & \text{for } 1 \leq x < 5 \\
      \frac{x-2}{4} & \text{for } 5 \leq x \leq 6 \\
      1 & \text{for } x > 6
      \end{cases}

   We must solve :math:`F_X(x_{0.85}) = 0.85` for :math:`x_{0.85}`.

   Which piece of the CDF should we use to replace the general expression :math:`F_x(x_{0.85})`?
   Since :math:`F_X(5) = \frac{3}{4} = 0.75 < 0.85`, we know that the 85th percentile has to be
   strictly greater than :math:`5`. Using Region 4, 

   .. math::
      &\frac{x_{0.85} - 2}{4} = 0.85\\
      &x_{0.85} - 2 = 3.4\\
      &x_{0.85} = 5.4

More examples
-----------------------------------------------------------

.. raw:: html

   <div class="video-placeholder" role="group" aria-labelledby="video-ch6-3-2">
     <iframe
       id="video-ch6-3-2"
       title="STAT 350 – Chapter 6.3.2 Continuous Random Variable Example 2 Video"
       src="https://www.youtube.com/embed/G-u5vHtQI3s?list=PLHKEwTHXfbagA3ybKLAcEJriGT-6k89c6"
       allowfullscreen>
     </iframe>
   </div>

Bringing It All Together
---------------------------

.. admonition:: Key Takeaways 📝
   :class: important

   1. The **cumulative distribution function** :math:`F_X(x) = P(X \leq x)` provides a universal framework for 
      both discrete and continuous random variables. It **eliminates the need for repeated integration** 
      when computing probabilities for continuous random variables.
   
   2. For continuous variables, **PDFs and CDFs are related** through differentiation and integration: 
      :math:`f_X(x) = F_X'(x)` and :math:`F_X(x) = \int_{-\infty}^x f_X(t) \, dt`.
   
   3. **Valid CDFs** must be non-decreasing, approach 0 as :math:`x \to -\infty`, approach 1 as :math:`x \to +\infty`, 
      and be right-continuous.
   
   4. **Probability calculations** become simpler with CDFS: :math:`P(X \leq a) = F_X(a)`, 
      :math:`P(X > a) = 1 - F_X(a)`, and :math:`P(a < X \leq b) = F_X(b) - F_X(a)`.
   
   5. **Piecewise CDFs** require careful attention—each region must 
      include all probability accumulated from previous regions.
   
   6. **Percentiles** are found by solving :math:`F_X(x_p) = p`.

In our next sections, we'll see how these CDF concepts apply to specific named distributions, 
starting with the most important continuous distribution in all of statistics: the normal distribution.

Exercises
~~~~~~~~~~~~~

1. **CDF Construction**: Given the PDF :math:`f_X(x) = 2x` for :math:`0 \leq x \leq 1`, 0 elsewhere:
   
   a) Find the CDF :math:`F_X(x)`
   b) Verify that your CDF satisfies all required properties
   c) Use the CDF to find :math:`P(0.2 < X \leq 0.8)`

2. **Piecewise CDF**: Consider the PDF:
   
   .. math::
   
      f_X(x) = \begin{cases}
      kx & \text{for } 0 \leq x \leq 2 \\
      k & \text{for } 2 < x \leq 4 \\
      0 & \text{elsewhere}
      \end{cases}
   
   a) Find the value of :math:`k` that makes this a valid PDF.
   b) Construct the complete CDF.
   c) Find :math:`P(1 < X < 3)` using the CDF.

3. **CDF Validation**: Determine whether the following function could be a valid CDF:
   
   .. math::
   
      F(x) = \begin{cases}
      0 & \text{for } x < 0 \\
      x^2 & \text{for } 0 \leq x < 1 \\
      1 & \text{for } x \geq 1
      \end{cases}
   
   If valid, find the corresponding PDF.

4. **Percentile Calculations**: For the CDF from Exercise 2,
   
   a) Find the median.
   b) Find the 75th percentile.
   c) Find the interquartile range.

5. **PDF-CDF Relationship**: Given :math:`F_X(x) = 1 - e^{-x}` for :math:`x \geq 0`, 0 elsewhere,
   
   a) Find the PDF :math:`f_X(x)`.
   b) Calculate :math:`P(X > 2)`.
   c) Find the value :math:`x` such that :math:`P(X \leq x) = 0.95`.