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Expectation of a Function of a Random Variable

Understanding Expectation of a Function g(X)

We often want to calculate the expected value of a function applied to a discrete random variable, written as \(E[g(X)]\), where:

  • \(X\) is a discrete random variable with known probability mass function (pmf) \(p(x)\)
  • g\((X)\) is any real-valued function defined on the values taken by \(X\)

Key Principle

The expectation of \(g(X)\) is calculated as a weighted average of \(g(x)\), weighted by the probability of each corresponding \(x\):

\[ E[g(X)]= \sum_{x} g(x) \cdot p(x) \]

This means:

If \(X\) takes values \(x_1, x_2, \ldots\) with probabilities \(p(x_1), p(x_2), \ldots\), then:

\[ E[g(X)]=\sum_i g(x_i) \cdot p(x_i) \]

Example

Let \(X\) take values \(-1, 0, and 1\) with:

  • P\((X = -1) = 0.2\)
  • \(P(X = 0) = 0.5\)
  • \(P(X = 1) = 0.3\)

We want to compute \(E[X²]\).

Let \(g(X) = X²\). Then:

  • \(g(-1) = 1\)
  • \(g(0) = 0\)
  • \(g(1) = 1\)

Now use the expectation formula:

\[ E[X^2] = 1 \cdot 0.2 + 0 \cdot 0.5 + 1 \cdot 0.3 = 0.5 \]

Application: Maximizing Expected Profit

Scenario: A store stocks a seasonal product.

  • Selling a unit earns \(b\) dollars.
  • Unsold units incur a loss of \(ℓ\) dollars per item.

Let \(X\) = number of units ordered by customers (a random variable with pmf \(p(i)\))

Let \(s\) = number of units stocked

Then the expected profit \(E[P(s)]\) is computed by breaking it into two cases:

  • If demand ≤ stock:
\[ P(s) = bX - (s - X)\ell \]

  • If demand > stock:

    \[ P(s) = sb \]

Combining these:

\[ E[P(s)] = \sum_{i=0}^s [bi - (s - i)\ell]p(i) + \sum_{i=s+1}^\infty sb \cdot p(i) \]

To find the optimal stocking level \(s*\), compute the value of \(s\) that satisfies:

\[ \sum_{i=0}^s p(i) < \frac{b}{b + \ell} \]

Utility Theory (Decision Under Uncertainty)

Suppose you must choose between two actions. Each leads to one of several consequences \(C_1, C_2, \ldots, C_n\), each with associated probabilities \(p_i\) (action 1) and \(q_i\) (action 2).

We define the utility \(u(C_i)\) of consequence \(C_i\) as the value uu for which the decision-maker is indifferent between:

  • receiving \(C_i\), or
  • taking a gamble with outcome:
    • best consequence CC with probability uu
    • worst consequence cc with probability \(1 - u\)

Then the expected utility of an action is:

\[ \sum_{i=1}^n p_i \cdot u(C_i) \]

Choose the action with the higher expected utility.

Linear Property of Expectation (Corollary)

If a and b are constants:

\[ E[aX + b] = aE[X] + b \]

This linearity property is useful in many calculations.

Moments

  • Mean (First Moment): \(E[X]\)
  • nth Moment: \(E[X^n] = \sum_x x^n \cdot p(x)\)

These moments are used to understand the shape and variability of a distribution.