Econ 120A Discrete Probability Distributions

Discrete Probability Distributions

Mean and Variance

• Population Mean ($\mu$):
  • Formula: $\mu = \sum x p(x)$
  • This is the average value of the random variable $X$, weighted by the probabilities of each value.
• Population Variance ($\sigma^2$):
  • Formula: $\sigma^2 = \sum (x - \mu)^2 p(x)$
  • This measures the spread or dispersion of the random variable $X$ around its mean.

Example: Errors in Economics Textbooks

• $X$ = number of errors per page in Economics textbooks

Variance Decomposition

• Formula:
  • $\sigma^2 = \sum (x - \mu)^2 p(x)$
  • Expanding the square:
    $\sigma^2 = \sum (x^2 - 2\mu x + \mu^2) p(x)$
  • Distributing $p(x)$:
    $\sigma^2 = \sum x^2 p(x) - 2\mu \sum x p(x) + \mu^2 \sum p(x)$
  • Since $\mu = \sum x p(x)$ and $\sum p(x) = 1$:
    $\sigma^2 = \sum x^2 p(x) - 2\mu^2 + \mu^2$
  • Simplified formula:
    $\sigma^2 = \sum x^2 p(x) - \mu^2$

Example Calculation

• Using the errors in economics textbooks example:
  • Var(X) = 0.25 – (0.21)2 = 0.2059

Expectations Operator

• E(.) Operator:
  • Takes the weighted average of (.), where the weights are the probabilities.
  • $E(X) = \sum x p(x) = \mu_X$
• Expected Value:
  • The expected value of a random variable is the weighted average of its possible values, weighted by their probabilities of occurring.
  • Note that this is the same as the mean.
  • The expected value of random variable X is its mean.

Expected Value of Functions of Random Variables

• If $R = g(X)$, then
  • $E(R) = E[g(X)] = \sum g(x) p(x) = \mu_R$
  • The expected value of random variable $R$ (a function of the random variable $X$) is also its mean.

Example Continued

• Calculating $E(X^2)$
  • Given $X = {x<em>1, x</em>2, \ldots, x_n}$
  • $E(X^2) = \sum x^2 p(x)$
  • Note that $E(X^2) = 0.25 \neq [E(X)]^2 = 0.21^2$

Expected Value of Squared Deviations from the Mean

• If $g(X) = (X - \mu_X)^2$
  • $E[g(X)] = \sum g(x) p(x) = \sum (x - \mu<em>X)^2 p(x) = \sigma</em>X^2$
  • Thus, $E[(X - \mu<em>X)^2] = \sigma</em>X^2$
  • The expected value of the squared deviation from the mean of the random variable $X$ is its variance, just like the expected value of $X$ is its mean.

Useful Rules for Expectations

• If $c$ is any constant:
  1. $E(c) = c$
  2. $E(X + c) = E(X) + E(c) = E(X) + c$
  3. $E(cX) = c E(X)$

Alternative Expression of Variance

• Using the expectations operator:
  • $\sigma<em>X^2 = E[(X - \mu</em>X)^2] = E[X^2 - 2X\mu<em>X + \mu</em>X^2]$
  • Applying linearity of expectation:
    $= E[X^2] + E[-2X\mu<em>X] + E[\mu</em>X^2]$
  • $= E[X^2] - 2\mu<em>X E[X] + E[\mu</em>X^2]$
  • $= E[X^2] - 2\mu<em>X \mu</em>X + \mu_X^2$
  • $= E[X^2] - \mu_X^2$
• Thus,
  • $\sigma<em>X^2 = E[(X - \mu</em>X)^2] = E[X^2] - \mu_X^2$

Summary

• Population mean of random variable $X$:
  • $\mu_X = \sum x p(x) = E(X)$
• Population variance of random variable $X$:
  • $\sigma<em>X^2 = \sum (x - \mu</em>X)^2 p(x) = \sum x^2 p(x) - \mu_X^2$
  • $= E[(X - \mu<em>X)^2] = E[X^2] - \mu</em>X^2$

Useful Rules (Revisited)

• If $c$ is any constant:
  1. $E(c) = c$
  2. $E(X + c) = E(X) + E(c)$
  3. $E(cX) = c E(X)$
  4. $Var(X + c) = Var(X)$
  5. $Var(cX) = c^2 Var(X)$

Rules Explained

1. Expected value of a constant ($c$):
   • $E[c] = c$
2. Expected value of the sum ($X + c$):
   • $E[X + c] = \sum (x + c) p(x) = \sum x p(x) + \sum c p(x) = E[X] + c \sum p(x)$
   • Since $\sum p(x) = 1$:
     $E[X + c] = E[X] + c = \mu_X + c$
3. Expected value of the product ($cX$):
   • $E[cX] = \sum c x p(x) = c \sum x p(x) = c E[X] = c \mu_X$
4. Variance of the sum ($X + c$):
   • $Var(X + c) = \sum ((x + c) - (\mu<em>X + c))^2 p(x) = \sum (x - \mu</em>X)^2 p(x) = Var(X)$
5. Variance of the product ($cX$):
   • $Var(cX) = \sum (cx - c \mu<em>X)^2 p(x) = c^2 \sum (x - \mu</em>X)^2 p(x) = c^2 Var(X)$