Find the expected value (EV) for the Random variable (X) by using this expected value calculator. It functions to estimate the probable average outcome or value of a random variable based on different possible outcomes. Also, you can get step-by-step calculations for the probability distribution.

## What is the Expected Value?

Expected value is the arithmetic mean or average value of a random variable based on all the various possible outcomes that occur frequently.

In probability and statistics, the expected value calculator is also known as the expectation calculator.

**For Example:**

Think of it like flipping a coin, there is a 50% chance you will get heads and a 50% chance you will get tails. The expected value is not exactly heads or tails, but somewhere in between.

In this case, the expected value would be 0.5 according to the formula which we have discussed below.

## Expected Value Formula

\( E(X) = \mu_x = x_{1}P(x_1) + x_{2}P(x_2) + … + x_{n}P(x_n) \)

By using the summation sign, the above equation can be rewritten as:

\( E(X) = \mu_x = \sum_{i=1}^{n} x_i * P(x_i) \)

Where,

- \(E(X)\): Represents the expected value of the random variable X
- \(\mu_x\): Indicates the mean of X
- \(\sum\): Symbol for summation
- \(P(x_i)\): Represents the probability of the value \((x_i)\)
- \(n\): The number of all possible outcomes
- \(x_i\): Referred to as the \(i^{th}\) outcome of the random variable X
- \(i\): Indicates the possible outcome of the random variable X

## How do you Calculate the Expected Value?

**Example:**

A dice has six sides, and each side has a number like 1, 2, 3, 4, 5, or 6.

Now, let's say you roll this dice. What number will you get? Because each number has an equal chance of showing up.

Let's calculate it:

The possible outcomes are the numbers from 1 to 6.

The probability of getting any single number is 1/6 because there are six sides on the dice.

Now, let's find the expected value using the formula:

Expected Value E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6)

Expected Value E(X) = 21/6 = 3.5

### Table E(X):

Outcome (X) | Probability P(X) | Weighted Sum: \(x_i * P(x_i)\) |
---|---|---|

1 | 1/6 | 1/6 |

2 | 1/6 | 2/6 |

3 | 1/6 | 3/6 |

4 | 1/6 | 4/6 |

5 | 1/6 | 5/6 |

6 | 1/6 | 6/6 |

Total |
1 |
21/6 |

Expected Value E(X) |
3.5 |

So, when you roll the dice many times, you can expect the average value to be around 3.5. You can even check it right by adding the same values into our expected value calculator.

## Steps to Use this Calculator:

**Step 1:** Enter the values for the probability of P(X) and the values of variable X in the designated boxes.

**Step 2:** Click Calculate

**Step 3:** Finally, this expected value calculator provides the E (X) Expected Value table along with step-by-step calculations.

## FAQs

### Can the Expected Value Be Negative?

**Yes, the expected value can be negative.** Consider a scenario where you are playing a game with two possible outcomes: winning or losing money. Let's say there's a 60% chance of gaining $10 and a 40% chance of losing $15.

In this case, the expected value would be:

- Expected Value = 0.60 × 10 + 0.40 × (− 15) = −1

This negative expected value means that, on average, you can expect to lose $1 per game due to the probabilities and values associated with each outcome.

### What Does it Mean If the Expected Value is Zero?

If the expected value is zero, it indicates that the average outcome of a random variable is equal to zero. In other words, the sum of the products of each possible outcome and its probability is equal to zero. This means that the positive and negative outcomes balance out which results in no gain or loss over a large number of trials.