Probability 2 Edits #3

_quarto.yml

@@ -33,7 +33,7 @@ book:
         - case_study_HCE/case_study_HCE.qmd
         - cv_regularization/cv_reg.qmd
         - probability_1/probability_1.qmd
-        # - probability_2/probability_2.qmd
+        - probability_2/probability_2.qmd
         # - inference_causality/inference_causality.qmd
         # - case_study_climate/case_study_climate.qmd
         # - sql_I/sql_I.qmd

probability_2/probability_2.qmd

@@ -220,7 +220,7 @@ $$\hat{Y}(x) = f_{\hat{\theta}}(x) = \hat{\theta_0} + \sum_{j=1}^p \hat{\theta_j
 In our estimating equation above, our sample $\Bbb{X}$, $\Bbb{Y}$ are random (often due to human error). Hence, the estimates we calculate from our samples $\hat{\theta}$ are also random, so our predictor $\hat{Y}(x)$ is also random. 
 :::
 
-Now taking a look at our original equations, we can see that they both have differing sources of randomness. For our observed relationship, $Y = g(x) + \epsilon$, $\epsilon$ represents measurement errors and reflects randomness from the future. For the estimation model, the data we have is a random sample collected from the population, so the randomness from the past.
+Now taking a look at our original equations, we can see that they both have differing sources of randomness. For our observed relationship, $Y = g(x) + \epsilon$, $\epsilon$ represents errors which occur during or after the observation or measurement process. For the estimation model, the data we have is a random sample collected from the population, which was constructed from decisions made before the measurement process.
 
 ## Bias-Variance Tradeoff {#sec-bias-variance-tradeoff}