CH10 - Expected Values of Discrete Random Variables

Just a practice worksheet for now, but remember:

Expected Value of a Discrete RV

$$E(X)=\sum _{j=1}^n{x}_j{P}_x(x_j)$$

Properties of Expected Value

1. For any constant c:

   $E(c)=c$

2. For a constant c and random variable X:

   $E(cX)=cE(X)$

3. For random variables X and Y:

   $E(X+Y)=E(X)+E(Y)$

---

Examples

Example 1: Fair Dice Roll

For a fair six-sided die, X = outcome of roll:

$E(X)=1(\frac{1}{6})+2(\frac{1}{6})+3(\frac{1}{6})+4(\frac{1}{6})+5(\frac{1}{6})+6(\frac{1}{6})=3.5$

Example 2: Binomial Random Variable

For a binomial random variable with n trials and probability p:

$E(X)=np$

---

Practice Problems

1. A game costs $5toplay.Youwin$10 with probability 0.4 and nothing with probability 0.6. Find the expected value of your winnings.

2. In a lottery, the probability of winning is 0.001. The prize is $1000andtheticketcosts$2. Calculate the expected value of playing.

---

In Summary

• Expected value represents the long-term average of a random variable
• It does not necessarily equal a possible value of the random variable
• Useful for decision making and risk assessment
• Forms the basis for more advanced probability concepts