In statistics, the expected value of a random variable is a measure of the central tendency of its probability distribution. In simple terms, it gives you an idea of what value you should expect to see on average over a large number of trials. Calculating the expected value can help in making predictions and decisions based on probabilities. In this article, we will discuss three ways to calculate an expected value: using a discrete probability distribution, a continuous probability distribution, and a decision tree.

1. Discrete Probability Distribution

In a discrete probability distribution, there are a finite or countably infinite number of possible outcomes (e.g., rolling a die). To calculate the expected value in this case, follow these steps:

a. List all possible outcomes and their corresponding probabilities.

b. Multiply each outcome by its probability.

c. Sum the products obtained in step b.

The result is the expected value for the discrete probability distribution.

Example: A simple dice roll:

Outcomes: 1, 2, 3, 4, 5, 6

Probabilities: Each outcome has a probability of 1/6

Expected Value = (1 × 1/6) + (2 × 1/6) + (3 × 1/6) + (4 × 1/6) + (5 × 1/6) + (6 × 1/6) = 3.5

2. Continuous Probability Distribution

In a continuous probability distribution, there are infinitely many possible outcomes within a specified range (e.g., height or weight measurements). To calculate the expected value in this case, follow these steps:

a. Define the range of possible outcomes and their corresponding probability density function (pdf).

b. Multiply each outcome by its probability density function.

c. Integrate the product obtained in step b over the entire range of possible outcomes.

The result is the expected value for the continuous probability distribution.

Example: A uniform distribution between 0 and 1:

Pdf: f(x) = 1 for x in [0,1]

Expected Value = Integral of x * f(x) over [0,1] = Integral of x over [0,1] = (1/2)(1)^2 – (1/2)(0)^2 = 0.5

3. Decision Tree

A decision tree is a visual representation of a decision-making process that can be used to calculate expected values in scenarios with multiple decisions and events. To calculate the expected value using a decision tree, follow these steps:

a. Draw the decision tree, including all possible decision nodes, event nodes, outcomes, and their corresponding probabilities.

b. Calculate the expected value at each event node by multiplying the outcomes by their probabilities and summing the products.

c. Move backward through the tree to find the expected values at decision nodes by choosing the highest (or lowest) expected value among the connected event nodes.

d. Identify the path with the greatest (or smallest) expected value.

The result is the optimal decision path with its expected value.

Example: A simple business investment decision:

Decision Node: Invest or Don’t Invest

Event Nodes for investing: Gain Profit (Probability = 60%) or Loss (Probability = 40%)

Outcomes: Gain $10,000 from profit or Lose $5,000 from loss

Expected Value for Gain: ($10,000 × 60%) + ($5,000 × 40%) = $8,000

Expected Value for Not Investing: $0

Best Decision Path: Invest with an Expected Value of $8,000