typos chapter 4

Author: cebe

Chapter4_TheGreatestTheoremNeverTold/LawOfLargeNumbers.ipynb

@@ -17,7 +17,7 @@
       "##The greatest theorem never told\n",
       "\n",
       "\n",
-      "This chapter focuses on an idea that is always bouncing around our minds, but is rarely made explicit outside books devoted to statistics. In fact, we've been used this simple idea in every example thus far. "
+      "This chapter focuses on an idea that is always bouncing around our minds, but is rarely made explicit outside books devoted to statistics. In fact, we've been using this simple idea in every example thus far. "
      ]
     },
     {
@@ -136,7 +136,7 @@
      "source": [
       "Looking at the above plot, it is clear that when the sample size is small, there is greater variation in the average (compare how *jagged and jumpy* the average is initially, then *smooths* out). All three paths *approach* the value 4.5, but just flirt with it as $N$ gets large. Mathematicians and statistician have another name for *flirting*: convergence. \n",
       "\n",
-      "Another very relevant question we can ask is *how quickly am I converging to the expected value?* Let's plot something new. For a specific $N$, let's do the above trials thousands of times and compute how far away we are from the true expected value, on average. But wait — *compute on average*? This simply the law of large numbers again! For example, we are interested in, for a specific $N$, the quantity:\n",
+      "Another very relevant question we can ask is *how quickly am I converging to the expected value?* Let's plot something new. For a specific $N$, let's do the above trials thousands of times and compute how far away we are from the true expected value, on average. But wait — *compute on average*? This is simply the law of large numbers again! For example, we are interested in, for a specific $N$, the quantity:\n",
       "\n",
       "$$D(N) = \\sqrt{ \\;E\\left[\\;\\; \\left( \\frac{1}{N}\\sum_{i=1}^NZ_i  - 4.5 \\;\\right)^2 \\;\\;\\right] \\;\\;}$$\n",
       "\n",
@@ -273,7 +273,7 @@
       "\n",
       "When is enough enough? When can you stop drawing samples from the posterior? That is the practitioners decision, and also dependent on the variance of the samples (recall from above a high variance means the average will converge slower). \n",
       "\n",
-      "We also should understand when the Law of Large Numbers fails. As the name implies, and comparing the graphs above for small $N$, the Law is only true for large sample sizes. Without this, the asymptotic result is not reliable. Knowing in what situations the Law fails can give use *confidence in how unconfident we should be*. The next section deals with this issue."
+      "We also should understand when the Law of Large Numbers fails. As the name implies, and comparing the graphs above for small $N$, the Law is only true for large sample sizes. Without this, the asymptotic result is not reliable. Knowing in what situations the Law fails can give us *confidence in how unconfident we should be*. The next section deals with this issue."
      ]
     },
     {