Expected Value Calculator

Our expected value calculator helps to find the probability expected value of a discrete random variable (X) and give you accurate results.

What is Expected Value?

In probability and statistics theory, the expected value is exactly what you might think it means intuitively: it is referred to as the return that you can expect for some kind of action, like how many multiple-choice questions you might get right if you guess on a multiple-choice test. The expected value of a random variable

\[ E(X) = E[X] = \sum_{i=1}^{n} x_i \cdot P(x_i) \]

,

uses probability to tell what outcomes to expect in the long run.

What is the Expected Value Formula?

Expected Value (EV) Formula:

\[ E(X) = \mu_x = x_1 P(x_1) + x_2 P(x_2) + \dots + x_n P(x_n) = \sum_{i=1}^{n} x_i \cdot P(x_i) \]

Where:

• E(X) = Expected value of the random variable X
• \(\mu_x\) = Mean of X
• \(\sum\) = Summation symbol
• P(x_i) = Probability of the outcome \(x_i\)
• x_i = The \(i^\text{th}\) outcome of the random variable X
• n = Number of possible outcomes
• i = Index of the outcome (from 1 to n)

How To Find Expected Value (Step-by-Step)

The formula is discussed earlier; here we have an example for a better understanding of the concept.

Example:

Example: Expected Value Calculation

Given numbers: \(X = 4, 8, 6, 3\) with corresponding probabilities \(P(X) = 0.1, 0.5, 0.04, 0.36\).

Expected Value Formula:

\[ E(X) = \mu_X = \sum_{i=1}^{n} x_i \cdot P(x_i) = x_1P(x_1) + x_2P(x_2) + \dots + x_nP(x_n) \]

Step 1: Substitute the values:

\[ E(X) = (4)(0.1) + (8)(0.5) + (6)(0.04) + (3)(0.36) \]

Step 2: Multiply each term:

\[ E(X) = 0.4 + 4 + 0.24 + 1.08 \]

Step 3: Sum the results:

\[ E(X) = 5.72 \]

Final Answer: The expected value is \(E(X) = 5.72\).

How Our Expected Value Calculator Works?

• Input the different “Outcomes” and their associated “Probabilities” in the respective fields
• Press the “Add Row” button if you have more values to generate new rows
• Press the “Calculate” button and that's all, you will get the expected value, expected value table and a step-by-step calculation

References:

From the authorized source of Wikipedia : Definition and formula
From the source of Investopedia : General understanding of EV