7.2: Central moments of probability density function

5. Calculating the Central Moment of pdf
   1. Expected value function \(E[x]\)
      1. Similar feature to indicate mean value for the variable
         1. What is the expected value of 3? \(E[3]\)
         2. How would you do that for a random variable \(x\)?
            • \(E[x]\) best guess would be the mean value
         3. Allows the argument to be multiplied by the pdf \[ E[f(x)] = \int_{-\infty}^{+\infty}f(x) p(x) dx \]
            • \(f(x)\) is the value person is adding up
            • \(p(x)\) is fraction of cases when \(f(x)\) occurs
      2. Expectation function is a linear operator:
         1. Distribution law applies: \(E[a + b]=E[a]+E[b]\)
         2. Multiply by a constant: \(E[7c] = 7E[c]\) when \(c\) is random variable
      3. Used to find the central moments of pdf \(\mu_{m}\) through \[ \mu_{m} = E[(x-x')^{m}] = \int_{-\infty}^{+\infty}(x-x')^{m} p(x) dx \] where \(f(x) = (x-x')^{m}\) as the argument of the Expected value function
   2. Zero\(^{\mathrm{th}}\) moment (\(m=0\)) \[ \mu_{0} = E[(x-x')^{0}] =\int_{-\infty}^{+\infty}(x-x')^{0} p(x) dx \] \[ \mu_{0} = E[1] = \int_{-\infty}^{+\infty}1 p(x) dx = 1 \]
      • Implies that area under pdf must have value of 1
   3. First moment (\(m=1\))
      1. Using expectation function first \[ \mu_{1} = E[(x-x')^{1}] = E[x] - E[x'] \] Which of those is constant? If expectation is like mean of random variable \[ \mu_{1} = E[x] -x' = \mathbf{0} \]
      2. Using in pdf form instead \[ \mu_{1} = \int_{-\infty}^{+\infty}(x-x')^{1} p(x) dx \] \[ \mu_{1} = \int_{-\infty}^{+\infty}x p(x) dx - x'\int_{-\infty}^{+\infty} p(x) dx = 0 \]
      3. In order for pdf version to work, must define true mean as \[ x' \equiv \int_{-\infty}^{+\infty}x p(x) dx \]
      4. If \(m=0\) defined area as 1, then \(x'\) acts as horizontal centroid of the area
   4. Second central moment (\(m=2\))
      1. Using the linearity of expectation function again \[ \mu_{2} = E[(x-x')^{2}] = E[x^{2}-2xx'+x'^{2}] \] \[\mu_{2} = E[x^{2}]-2x'E[x]+x'^{2} \] \[ \mu_{2} = E[x^{2}] - x'^{2} \equiv \sigma^{2} \] where \(\sigma^{2}\) is the variance of the random data
      2. Same expansion could be applied to pdf \[ \mu_{2} = \int_{-\infty}^{+\infty}(x-x')^{2} p(x) dx = \sigma^{2} \]
      3. Note the different terms:
         1. variance \(\sigma^{2}\) is square of units relative to the mean
         2. standard deviation \(\sigma\) is in the same units of random variable
         3. mean square is the non-centered square of the random variable \[ \psi^{2} = \int_{-\infty}^{+\infty}x^{2} p(x) dx = \sigma^{2} + x'^{2} \]
   5. Third central moment (\(m=3\)) \[ \mu_{3} = E[(x-x')^{3}] =\int_{-\infty}^{+\infty}(x-x')^{3} p(x) dx \]
      1. Defines the skewness or asymmetry of the pdf
      2. \(Sk = \frac{\mu_{3}}{\sigma^{3}}\) making it dimensionless and signed value
      3. Right shifted (tail stretches to the right) \(Sk>0\)
      4. Left shifted when \(Sk<0\)
      5. Balanced when \(Sk = 0\)
   6. Fourth central moment (\(m=4\)) \[ \mu_{4} = E[(x-x')^{4}] =\int_{-\infty}^{+\infty}(x-x')^{4} p(x) dx \]
      1. Defines the kurtosis or flatness versus peakedness of the pdf
      2. \(Ku = \frac{\mu_{4}}{\sigma^{4}}\) since scaled is again dimensionless
      3. \(Ku = 3\) for Bell curve (normal distribution; next section)
      4. \(Ku > 3\) when sharp top peak (high derivative over the top)
      5. \(0 < Ku < 3\) when more flat on top
   7. Drawn graphically as part of lecture
   8. All moments are independent of one another except for the true mean \(x'\) since that is embedded within \(m=\{2,3,4\}\)