3.4 Expectation

• Parameters extracted from datasets, such as mean and standard deviation can also be modelled via ideal versions using random variables.

• Definition: The expectation of a random variable $X$ is $\mathbb{E}[X] = \displaystyle{ \sum_{x\in X(\Omega)} } x \cdot p_X (x)$ .

• Expectation is the mean of the random variable $X$.

• $\mathbb{E}[X]$ does not necessarily match with some state, it’s more like a center of mass between all the states of $X$.

• Example: Game with reward after flipping a coin 3 times.

• Reward:

  • 0 usd, if there are 0 or 1 head.
  • 1 usd, if there are 2 heads.
  • 8 usd, if there are 3 heads.

• Model:

  • $X$: Number of heads after flipping a fair coin 3 times.
  • $Y$: Reward obtained from this game.

• Sample space: HHH, HHT, HTH, THH, THT, TTH, HTT, TTT .

• Values obtained:

  • $p_X(0) = 1/8, p_X(1) = 3/8, p_X(2) = 3/8, p_X(3) = 1/8$
  • $p_Y(0) = p_X(0) + p_X(1) = 4/8$
  • $p_Y(1) = p_X(2) = 3/8$
  • $p_Y(8) = p_X(3) = 1/8$

• Expected reward:

  • $E[Y] = 0\cdot 4/8 + 1\cdot 3/8 + 8\cdot 1/8 = 11/8$ .
  • Therefore, if the cost of the game is greater than $11/8$ usd, then, we would lose money (on average).