remove duplicate figures and add library command for UsingR

Author: smcclatchy

episodes/conditional-probabilities-expectations.Rmd

@@ -61,6 +61,7 @@ son, we would go with the average height:
 
 ```{r height_hist,message=FALSE,fig.cap="Histogram of son heights."}
 library(rafalib)
+library(UsingR)
 mypar(1, 1)
 data(father.son, package="UsingR")
 x=round(father.son$fheight) ## round to nearest inch
@@ -69,8 +70,6 @@ hist(y, breaks=seq(min(y), max(y)))
 abline(v=mean(y), col="red", lwd=2)
 ```
 
-![Histogram of son heights.](./fig/03-conditional-probabilities-expectations-height_hist-1.png)
-
 In this case, we can also approximate the distribution of $Y$ as normal, which 
 implies the mean maximizes the probability density. 
 
@@ -93,9 +92,6 @@ hist(y[x==71],
      main="", 
      xlim=range(y))
 ```
-
-![Son versus father height (left) with the red lines denoting the stratum defined by conditioning on fathers being 71 inches tall. Conditional distribution: son height distribution of stratum defined by 71 inch fathers.](./fig/03-conditional-probabilities-expectations-conditional_distribution-1.png)
-
 The best guess is still the expectation, but our strata has changed from all the
 data, to only the $Y$ with $X=71$. So we can stratify and take the average, 
 which is the conditional expectation. Our prediction for any $x$ is therefore:
@@ -135,8 +131,6 @@ hist(y[x==71],
 abline(v = fit$coef[1] + fit$coef[2]*71, col=1)
 ```
 
-![Son versus father height showing predicted heights based on regression line (left). Conditional distribution with vertical line representing regression prediction.](./fig/03-conditional-probabilities-expectations-regression-1.png)
-
 In this particular case, the regression line provides an optimal prediction 
 function for $Y$. But this is not generally true because, in the typical machine
 learning problems, the optimal $f(x)$ is rarely a simple line.