refine text on probability density functions and their properties.

probability/16_continuous_distribution.py

@@ -14,7 +14,7 @@
 
 import marimo
 
-__generated_with = "0.11.26"
+__generated_with = "0.12.6"
 app = marimo.App(width="medium")
 
 
@@ -26,7 +26,9 @@ def _(mo):
 
         _This notebook is a computational companion to ["Probability for Computer Scientists"](https://chrispiech.github.io/probabilityForComputerScientists/en/part2/continuous/), by Stanford professor Chris Piech._
 
-        So far, all the random variables we've explored have been discrete, taking on only specific values (usually integers). Now we'll move into the world of **continuous random variables**, which can take on any real number value. Continuous random variables are used to model measurements with arbitrary precision like height, weight, time, and many natural phenomena.
+        Continuous distributions are what we need when dealing with random variables that can take any value in a range, rather than just discrete values. 
+
+        The key difference here is that we work with probability density functions (PDFs) instead of probability mass functions (PMFs). It took me a while to really get this - the PDF at a point isn't actually a probability, but rather a density.
         """
     )
     return
@@ -38,20 +40,17 @@ def _(mo):
         r"""
         ## From Discrete to Continuous
 
-        To make the transition from discrete to continuous random variables, let's start with a thought experiment:
-
-        > Imagine you're running to catch a bus. You know you'll arrive at 2:15pm, but you don't know exactly when the bus will arrive. You want to model the bus arrival time (in minutes past 2pm) as a random variable $T$ so you can calculate the probability that you'll wait more than five minutes: $P(15 < T < 20)$.
+        Making the jump from discrete to continuous random variables requires a fundamental shift in thinking. Let me walk you through a thought experiment:
 
-        This immediately highlights a key difference from discrete distributions. For discrete distributions, we described the probability that a random variable takes on exact values. But this doesn't make sense for continuous values like time.
+        > You're rushing to catch a bus. You know you'll arrive at 2:15pm, but the bus arrival time is uncertain. If you model the bus arrival time (in minutes past 2pm) as a random variable $T$, how would you calculate the probability of waiting more than five minutes: $P(15 < T < 20)$?
 
-        For example:
+        This highlights a crucial difference from discrete distributions. With discrete distributions, we calculated probabilities for exact values, but this approach breaks down with continuous values like time.
 
+        Consider these questions:
         - What's the probability the bus arrives at exactly 2:17pm and 12.12333911102389234 seconds?
-        - What's the probability of a child being born weighing exactly 3.523112342234 kilograms?
+        - What's the probability a newborn weighs exactly 3.523112342234 kilograms?
 
-        These questions don't have meaningful answers because real-world measurements can have infinite precision. The probability of a continuous random variable taking on any specific exact value is actually zero!
-
-        ### Visualizing the Transition
+        These questions have no meaningful answers because continuous measurements can have infinite precision. In the continuous world, the probability of a random variable taking any specific exact value is actually zero!
 
         Let's visualize this transition from discrete to continuous:
         """
@@ -150,44 +149,43 @@ def _(mo):
         r"""
         ## Probability Density Functions
 
-        In the world of discrete random variables, we used **Probability Mass Functions (PMFs)** to describe the probability of a random variable taking on specific values. In the continuous world, we need a different approach.
+        While discrete random variables use Probability Mass Functions (PMFs), continuous random variables require a different approach — Probability Density Functions (PDFs).
 
-        For continuous random variables, we use a **Probability Density Function (PDF)** which defines the relative likelihood that a random variable takes on a particular value. We traditionally denote the PDF with the symbol $f$ and write it as:
+        A PDF defines the relative likelihood of a continuous random variable taking particular values. We typically denote this with $f$ and write it as:
 
         $$f(X=x) \quad \text{or simply} \quad f(x)$$
 
-        Where the lowercase $x$ implies that we're talking about the relative likelihood of a continuous random variable which is the uppercase $X$.
+        Where the lowercase $x$ represents a specific value our random variable $X$ might take.
 
         ### Key Properties of PDFs
 
-        A **Probability Density Function (PDF)** $f(x)$ for a continuous random variable $X$ has these key properties:
+        For a PDF $f(x)$ to be valid, it must satisfy these properties:
 
-        1. The probability that $X$ takes a value in the interval $[a, b]$ is:
+        1. The probability that $X$ falls within interval $[a, b]$ is:
 
            $$P(a \leq X \leq b) = \int_a^b f(x) \, dx$$
 
-        2. The PDF must be non-negative everywhere:
+        2. Non-negativity — the PDF can't be negative:
 
            $$f(x) \geq 0 \text{ for all } x$$
 
-        3. The total probability must sum to 1:
+        3. Total probability equals 1:
 
            $$\int_{-\infty}^{\infty} f(x) \, dx = 1$$
 
-        4. The probability that $X$ takes any specific exact value is 0:
+        4. The probability of any exact value is zero:
 
            $$P(X = a) = \int_a^a f(x) \, dx = 0$$
 
-        This last property highlights a key difference from discrete distributions: the probability of a continuous random variable taking on an exact value is always 0. Probabilities only make sense when talking about ranges of values.
-
-        ### Caution: Density ≠ Probability
+        This last property reveals a fundamental difference from discrete distributions — with continuous random variables, probabilities only make sense for ranges, not specific points.
 
-        A common misconception is to think of $f(x)$ as a probability. It is instead a **probability density**, representing probability per unit of $x$. The values of $f(x)$ can actually exceed 1, as long as the total area under the curve equals 1.
+        ### Important Distinction: Density ≠ Probability
 
-        The interpretation of $f(x)$ is only meaningful when:
+        One common mistake is interpreting $f(x)$ as a probability. It's actually a **density** — representing probability per unit of $x$. This is why $f(x)$ values can exceed 1, provided the total area under the curve equals 1.
 
-        1. We integrate over a range to get a probability, or
-        2. We compare densities at different points to determine relative likelihoods.
+        The true meaning of $f(x)$ emerges only when:
+        1. We integrate over a range to obtain an actual probability, or
+        2. We compare densities at different points to understand relative likelihoods.
         """
     )
     return
@@ -665,16 +663,18 @@ def _(fig_to_image, mo, np, plt, sympy):
     # Detailed calculations for our example
     _calculations = mo.md(
         f"""
-        ### Calculating Expectation and Variance for Our Example
+        ### Computing Expectation and Variance
 
-        Let's calculate the expectation and variance for the PDF:
+        > _Note:_ The following mathematical derivation is included as reference material. The credit for this approach belongs to ["Probability for Computer Scientists"](https://chrispiech.github.io/probabilityForComputerScientists/en/part2/continuous/) by Chris Piech.
+
+        Let's work through the calculations for our PDF:
 
         $$f(x) = \\begin{{cases}} 
         \\frac{{3}}{{8}}(4x - 2x^2) & \\text{{when }} 0 < x < 2 \\\\ 
         0 & \\text{{otherwise}} 
         \\end{{cases}}$$
 
-        #### Expectation Calculation
+        #### Finding the Expectation
 
         $$E[X] = \\int_{{-\\infty}}^{{\\infty}} x \\cdot f(x) \\, dx = \\int_0^2 x \\cdot \\frac{{3}}{{8}}(4x - 2x^2) \\, dx$$
 
@@ -684,9 +684,9 @@ def _(fig_to_image, mo, np, plt, sympy):
 
         $$E[X] = \\frac{{3}}{{8}} \\cdot \\frac{{32 - 12}}{{3}} = \\frac{{3}}{{8}} \\cdot \\frac{{20}}{{3}} = \\frac{{20}}{{8}} = {E_X}$$
 
-        #### Variance Calculation
+        #### Computing the Variance
 
-        First, we need $E[X^2]$:
+        We first need $E[X^2]$:
 
         $$E[X^2] = \\int_{{-\\infty}}^{{\\infty}} x^2 \\cdot f(x) \\, dx = \\int_0^2 x^2 \\cdot \\frac{{3}}{{8}}(4x - 2x^2) \\, dx$$
 
@@ -696,11 +696,11 @@ def _(fig_to_image, mo, np, plt, sympy):
 
         $$E[X^2] = \\frac{{3}}{{8}} \\cdot \\frac{{20 - 64/5}}{{1}} = {E_X2}$$
 
-        Now we can calculate the variance:
+        Now we calculate variance using the formula $Var(X) = E[X^2] - (E[X])^2$:
 
         $$\\text{{Var}}(X) = E[X^2] - (E[X])^2 = {E_X2} - ({E_X})^2 = {Var_X}$$
 
-        Therefore, the standard deviation is $\\sqrt{{\\text{{Var}}(X)}} = {Std_X}$.
+        This gives us a standard deviation of $\\sqrt{{\\text{{Var}}(X)}} = {Std_X}$.
         """
     )
     mo.vstack([_img, _calculations])
@@ -779,7 +779,7 @@ def _(mo):
     return
 
 
-@app.cell
+@app.cell(hide_code=True)
 def _(mo):
     mo.md(r"""Appendix code (helper functions, variables, etc.):""")
     return
@@ -971,7 +971,6 @@ def _(np, plt, sympy):
         1. Total probability: ∫₀² {C}(4x - 2x²) dx = {total_prob}
         2. P(X > 1): ∫₁² {C}(4x - 2x²) dx = {prob_gt_1}
         """
-
     return create_example_pdf_visualization, symbolic_calculation