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docs/How does Maximum Likelihood Estimation work.rst

@@ -76,15 +76,31 @@ Exponential Log-PDF: :math:`ln(f(t)) = ln(\lambda)-\lambda t`
 
 Now we substitute in :math:`\lambda=0.1` and :math:`t = [27, 64, 3, 18, 8]`
 
+doing some equation testing here
+
 .. math::
 
     \begin{align}
-    L(\lambda=0.1|t=[27, 64, 3, 18, 8]) = (ln(0.1)-0.1 \times 27)\\
-    &\qquad + (ln(0.1)-0.1 \times 64) + (ln(0.1)-0.1 \times 3)\\
-    &\qquad + (ln(0.1)-0.1 \times 18) + (ln(0.1)-0.1 \times 8)\\
-    &\qquad = -23.512925
+    L(\lambda=0.1|t=[27, 64, 3, 18, 8])\\
+    = (ln(0.1)-0.1 \times 27) + (ln(0.1)-0.1 \times 64) + (ln(0.1)-0.1 \times 3)\\
+    + (ln(0.1)-0.1 \times 18) + (ln(0.1)-0.1 \times 8)\\
+    = -23.512925
     \end{align}
 
+.. math::
+
+    L(\lambda=0.1|t=[27, 64, 3, 18, 8])\\
+    = (ln(0.1)-0.1 \times 27) + (ln(0.1)-0.1 \times 64) + (ln(0.1)-0.1 \times 3)\\
+    + (ln(0.1)-0.1 \times 18) + (ln(0.1)-0.1 \times 8)\\
+    = -23.512925
+
+.. math::
+
+    L(\lambda=0.1|t=[27, 64, 3, 18, 8])
+    = (ln(0.1)-0.1 \times 27) + (ln(0.1)-0.1 \times 64) + (ln(0.1)-0.1 \times 3)
+    + (ln(0.1)-0.1 \times 18) + (ln(0.1)-0.1 \times 8)
+    = -23.512925
+
 Here's where the optimization part comes in. We need to vary :math:`\lambda` until we maximize the log-likelihood.
 The following graph shows how the log-likelihood varies as :math:`\lambda` varies.