什么是期望值？

Expectation, or expected value, is the mean of a random variable. It can be thought of as the average outcome that would be obtained after numerous trials of an experiment.

Suppose $X$ is a random variable from a sample space $\Omega$. Then the expected value of $X$ is $$\mathbb{E}[X] = \sum\limits_{x \in \Omega} Pr(x)X(x).$$

Some expected value problems require counting possible outcomes using permutations before their probabilities can be calculated.

An expected value can be thought of as a "weighted average," or the predicted value of a variable.

It is calculated as the sum of every possible value multiplied by the probability of that value occurring. Or, $$\sum\limits_{i=1}^np_i\cdot x_i$$ where $x_i$ is an event within the sample space, $p_i$ is the probability of that event occurring, and there are $n$ events total within that sample space.

For example, if we have a 4-sided dice with p(rolling a 1) = $\frac{1}{3}$, p(rolling a 2) = $\frac{1}{2}$, p(rolling a 3) = $\frac{3}{24}$, p(rolling a 4) = $\frac{1}{24}$, our expected value would be $$1\cdot\frac{1}{3}+2\cdot\frac{1}{2}+3\cdot\frac{3}{24}+4\cdot\frac{1}{24}=\frac{15}{8}.$$

To lead up to the concept of expected value, let's consider a few quick scenarios first.

Imagine you have a fairly weighted coin. When you flip heads, you earn $\$1$, but when you flip tails, you unfortunately don't earn any money.

Now let's spice things up a bit. This time, the coin's weighted so that it lands on heads $80\%$ of the time and tails $20\%$ of the time.

Now you've had a taste of how expected value works! Expected value is basically a weighted average of different possible outcomes, calculated by the formula $$E(X) = \sum\limits_{i=1}^{n}X_i P(X_i).$$ Let's break down this formula.

• $X$ denotes an event — in our example problem, our two events were (1) flipping heads and earning $\$1$, and (2) flipping tails and not earning anything.
• $P(X)$ denotes the probability of a particular event occurring — in the first question our probabilities were $50\%$ and $50\%$, but in the second question they were $80\%$ and $20\%$.
• The summation allows us to sum up the products of events and probabilities across all possible events.

You may have gotten the answer to the second question by making an educated guess, and that's okay! If we look at it using our formula, we find that $$E(X) = \sum\limits_{i=1}^{n}X_i P(X_i) = (\$1)(0.80) + (\$0)(0.20) = \boxed{\$0.80}.$$

Let's try a bit more involved of a question. Imagine you had a fairly weighted die. If you roll a $1$, $2$, or $3$, you win $\$2$. If you roll a $4$ or $5$, you win $\$6$. If you roll a $6$, you win $\$12$.

Now you've got expected value under your belt. Check out the problems in this topic if you want some practice, or if you want to go deeper in the world of probability, why not try some linearity of expectation?

Expected Value

The expected value in probability is a more complicated way of saying mean, which we are already pretty familiar with. More specifically, when talking about the expected value, we are talking about the weighted average.

In the real world, we can use expected value to what outcomes to expect in the long run, especially for uncertain outcomes. For example, we can use expected value when analyzing investment returns and when predicting lottery ticket wins.

To find the expected value of a discrete random variable, we want to sum the product of the random value and its probability.

Mathematically, to calculate the expected value of a discrete random variable, we use the following formula:

$$E(X) = \sum_{i} x_i \cdot P(X = x_i)$$

, where $E(X)$ is the expected value and $X$ is a random variable.

We can also find the expected value of continuous random variables and density functions, but that is slightly beyond the scope of this course. All you need to know is that you would use integration (calculus!) to find this value.

Overall, we will use expected value more and more when we calculate things in the long run or calculate values with a large number of trials. Think back to the law of large numbers: as the number of trials increases, the average outcomes will get closer and closer to the expected value.

Looking Further

In the next lessons, you will learn about the linearity of expectation and compound events. From there, we will move into independent and dependent events!

(Video) How to Calculate Expected Value

by The Organic Chemistry Tutor

In this video by The Organic Chemistry Tutor, you will learn how to calculate the expected value of winning a game.

The equation to calculate the expected value is: $(Xi*Pi )+ (X2*P2)$ where $X$ refers to the outcome and $P$ refers to the probability of that outcome.

In the example in this video, the key points to note are there is a 20% chance that Lisa wins, and if she does, she will win $500. Therefore, there must be an 80% chance that Lisa loses (since 100%-20% = 80%). We know that if she loses, she loses 100 dollars.

Let's plug this information into the expected value equation above:

Expected Value = $(500*0.20) + ((-100)*0.8)$ which equals $20$ dollars.

Keep watching the video to go through a second example!