Mean Discrete Random Variable- Calculating Expected Values
What Is a Discrete Random Variable?
A discrete random variable takes on specific, separate values. These aren't measurements that can be split into fractions. We're talking about counts—number of heads in 10 coin flips, number of emails arriving per hour, number of defective items in a batch.
The variable can only take values like 0, 1, 2, 3... never 1.5 or π. That's the deal with discrete.
The Expected Value (Mean) Explained
The expected value is the long-run average you'd see if you repeated an experiment infinite times. It's not a value you'll actually observe in a single trial. That's important to understand upfront.
If you roll a fair die 100,000 times and average the results, you'll get something close to 3.5. That's the expected value. No single die roll gives you 3.5, but the mean of many rolls converges to it.
The Formula
For a discrete random variable X with possible values x₁, x₂, x₃... xₙ and corresponding probabilities P(X = xᵢ), the expected value is:
E(X) = Σ [xᵢ × P(X = xᵢ)]
That's it. Multiply each outcome by its probability, then sum everything up.
Working Through a Real Example
Let's say a game costs $5 to play. You flip two fair coins. If both are heads, you win $20. Otherwise, you win nothing.
What's the expected value of your profit?
Step 1: Define the random variable
Let X = profit from playing once.
X = $20 - $5 = $15 when you win
X = $0 - $5 = -$5 when you lose
Step 2: Find the probabilities
P(both heads) = ¼ = 0.25
P(not both heads) = ¾ = 0.75
Step 3: Apply the formula
E(X) = (15 × 0.25) + (-5 × 0.75)
E(X) = 3.75 - 3.75
E(X) = $0
The game is fair. Over many plays, you'd break even on average.
Common Mistakes to Avoid
- Forgetting to include all outcomes. Probabilities must sum to 1. If they don't, you've missed something.
- Confusing the expected value with the most likely outcome. E(X) = 3.5 doesn't mean 3 or 4 is most likely for a die roll. It just means the average converges there.
- Ignoring the cost of entry. In gambling or business scenarios, always account for what you pay to play.
- Using the wrong formula for continuous variables. This method only works for discrete cases. Continuous variables require integration.
Properties of Expected Value
These rules make calculations easier when you're dealing with multiple variables:
- Linearity: E(aX + b) = aE(X) + b. You can pull constants out and add expected values together.
- Addition: E(X + Y) = E(X) + E(Y). This holds regardless of whether X and Y are independent.
- Multiplication: E(XY) = E(X) × E(Y) only if X and Y are independent. Don't assume this without checking.
Expected Value vs. Variance
Don't confuse these. Expected value measures the center of the distribution. Variance measures how spread out the values are around that center.
Two games might both have E(X) = $10, but one could have you winning $10 every time while the other has you oscillating between $0 and $20. The variance tells you which.
Comparison: Expected Value Calculations
| Scenario | Values (x) | Probabilities P(x) | E(X) Calculation | Result |
|---|---|---|---|---|
| Fair coin flip | 0, 1 | 0.5, 0.5 | (0×0.5)+(1×0.5) | 0.5 |
| Loaded die | 1, 2, 3, 4, 5, 6 | 0.1, 0.1, 0.1, 0.1, 0.1, 0.5 | Σ(x × P(x)) | 4.5 |
| Investment A | -100, 200 | 0.4, 0.6 | (-100×0.4)+(200×0.6) | 80 |
| Investment B | -50, 100 | 0.3, 0.7 | (-50×0.3)+(100×0.7) | 55 |
When Expected Value Lies to You
Consider a lottery: 1 in 300 million chance to win $100 million. Ticket costs $2.
E(X) = (-2 × 0.999999997) + (99999998 × 0.000000003) ≈ -$1.67
Technically positive expected value doesn't exist here. But even if it did, the variance is astronomical. You'd need to play millions of times before converging to the mean. Nobody does that.
Expected value is a theoretical tool. It doesn't tell you what happens in practice with limited trials.
How to Calculate Expected Value: Step-by-Step
- List all possible outcomes. Write down every value your random variable can take.
- Assign probabilities. Each outcome gets a probability between 0 and 1. They must sum to exactly 1.
- Multiply each outcome by its probability. This gives you the weighted contribution of each outcome.
- Sum all the products. That's your expected value.
- Interpret the result. Remember—this is a long-run average, not a guaranteed outcome.
Practical Applications
Expected value shows up everywhere once you know what to look for:
- Insurance: Companies price policies using expected medical costs versus premium collected.
- Casinos: Every game is designed so E(profit for house) > 0. That's how they stay in business.
- Business decisions: Comparing expected profits of different strategies before committing resources.
- Quality control: Estimating average defects per unit in manufacturing.
The Bottom Line
Expected value is straightforward arithmetic: multiply outcomes by probabilities, then add. The math isn't hard. The hard part is correctly identifying what outcomes and probabilities actually apply to your situation.
Most errors come from incomplete lists of outcomes, incorrect probability assignments, or forgetting to include costs and benefits that affect the final values. Get those right, and the calculation takes care of itself.