auto-update(nvim): 2025-02-12 13:22:06

documents/by-course/pstat-120a/course-notes/main.typ

@@ -1665,6 +1665,61 @@ $F_z(x)$, we use the special $phi(x)$ and $Phi(x)$.
   - use the standard normal probability table in the textbook
 ]
 
+To evaluate negative values, we can use the symmetry of the normal distribution
+to apply the following identity:
+
+$
+  Phi(-x) = 1 - Phi(x)
+$
+
+== General normal distributions
+
+The general family of normal distributions is obtained by linear or affine
+transformations of $Z$. Let $mu$ be real, and $sigma > 0$, then
+
+$
+  X = sigma Z + mu
+$
+is also a normally distributed random variable with parameters $(mu, sigma^2)$.
+The CDF of $X$ in terms of $Phi(dot)$ can be expressed as
+
+$
+  F_X (x) &= P(X <= x) \
+  &= P(sigma Z + mu <= x) \
+  &= P(Z <= (x - mu) / sigma) \
+  &= Phi((x-mu)/sigma)
+$
+
+Also,
+
+$
+  f(x) = F'(x) = dif / (dif x) [Phi((x-u)/sigma)] = 1 / sigma phi((x-u)/sigma) = 1 / sqrt(2 pi sigma^2) e^(-((x-mu)^2) / (2sigma^2))
+$
+
+#definition[
+  Let $mu$ be real and $sigma > 0$. A random variable $X$ has the _normal distribution_ with mean $mu$ and variance $sigma^2$ if $X$ has density function
+
+  $
+    f(x) = 1 / sqrt(2 pi sigma^2) e^(-((x-mu)^2) / (2sigma^2))
+  $
+
+  on the real line. Abbreviate this by $X ~ N(mu, sigma^2)$.
+]
+
+#fact[
+  Let $X ~ N(mu, sigma^2)$ and $Y = a X + b$. Then
+  $
+    Y ~ N(a mu + b, a^2 sigma^2)
+  $
+
+  That is, $Y$ is normally distributed with parameters $(a mu + b, a^2 sigma^2)$.
+  In particular,
+  $
+    Z = (X - mu) / sigma ~ N(0,1)
+  $
+  is a standard normal variable.
+]
+
 = Lecture #datetime(day: 11, year: 2025, month: 2).display()
 
 == Expectation