Expected value

Imagine this: You've always dreamed of starting your own business, and you've decided to sell footwear. But now comes the question, which brand will you work with? Nike or Adidas? As you weigh the pros and cons of each, you come to understand that what matters is what is the expected profit of each choice. Understanding the expected value of your potential earnings could be the key to making the right decision for your startup. Expected value is the cool kid of probability and statistics and for a good reason. It is one of the easiest and most useful tools to work with, It is used to estimate the average outcome of a random event in long-term behavior like the expected profit from a startup.

In machine learning, decision-making, risk management, and finance, expected value works wonders by helping us study complex systems and take the best action. With its ability to provide insight into potential outcomes, the expected value is an essential tool for anyone looking to make informed decisions and mitigate risks. In this topic, you will learn what expected value is, how to calculate it, and how can it be used for decision-making.

Probability distribution

probability distribution is a function that tells us how likely different things are to happen. the way to get it is by listing all possible outcomes from a random variable and corresponding probabilities.

For Example, if we are rolling a fair six-sided dice. Each side has the same probability of $1\over6$ to happen.

So, its probability distribution will be as the following.

However, making your probability distribution is your first step in finding the Expected value.

Calculate the expected value

The expected value of a random variable is determined by multiplying each possible outcome by its corresponding probability and summing all these products.

$$E(X)= \sum_{i=1}^k {p_i}{x_i} ={p_1}{x_1}+{p_2}{x_2}+{p_3}{x_3}+...+{p_k}{x_k}$$

And here it is, this formula gives us an estimation of the average value we can expect to obtain from an event over the long run.

Example: Expected value of a fair dice.

For a fair dice with six faces, the outcome is assigned to a random variable $X$, as each face has the same probability of ${1\over6}$ to come up so:

$E(X)=1*{1\over6}+2*{1\over6}+3*{1\over6}+4*{1\over6}+5*{1\over6}+6*{1\over6}=3.5$

So, if we rolled this dice many times and took the average of the outcomes, we expect the average to be approximately 3.5.

Example: Sum of two dice.

Let's conduct the same experiment, but this time we will focus on the sum of the two-dice roll. The experiment will yield a total of $6 × 6 = 36$ possible pairs of outcomes, The smallest possible sum is 2, while the largest sum is 12. This figure shows all the possible outcomes.

Upon examining the figure, it is evident that the sum of 2 appears in only one combination. Therefore, the probability of this outcome can be calculated by dividing the number of appearances of this event by the total number of outcomes. Hence, the probability of the sum of 2 is $1\over36$. The same principle applies to the sum of 9, resulting in a probability of $4\over36$.

The following probability distribution shows the probabilities for each sum.

The expected value will be calculated as follows,

$E(x)=2*{1\over36}+3*{2\over36}+4*{3\over36}+5*{4\over36}+6*{5\over36}+7*{6\over36}+8*{5\over36}+9*{4\over36}+10*{3\over36}+11*{2\over36}+12*{1\over36}=7$

Properties of expected value

There are some properties of expected values that make it easier in calculations and these are the most important properties.

1. Expected value of a constant

$E(a)=a$

The expected value of a constant is the constant itself.

2. Linearity of expected value

$E[X1 + X2 + ... + Xn] = E[X1] + E[X2] + ... + E[Xn]$

The expected value of the sum of independent random variables is equal to the sum of their expected values. Let's take an example to understand this:

Example: Monthly Income Expectation.

A businessman who owns two companies found that the expected monthly income for the first company is $50,000, and for the second company is$73,000. To find the total expected monthly income from the two companies we use the linearity property:

$E(company1+company2)=E(company1)+E(company2) = 50,000+73,000=123,000$$

However, let's recall the example of the two dice. Now, we have a simpler method to compute its expected value:

$E(dice1+dice2)=E(dice1)+E(dice2) =3.5 + 3.5 =7$

3. Product of expected value

$E[X * Y] = E[X] * E[Y]$

The expected value of a product of two independent random variables is equal to the product of its expected values.

Example: Expected daily revenue

What is the expected daily revenue if the expected number of products sold per day is 50, and the expected price of each product is $10? Let's call the number of products by $X$ and the price of each product by $Y$. Now, to calculate the revenue we multiply the number of sold products by its price.

$E(X*Y)=E(X)*E(Y)= 50 * 10=500$ $ of revenue.

Conclusion

Expected value is both a very useful and simple tool with numerous applications. It provides a single numerical value that represents the average outcome of a random variable, making it useful for decision-making, it works by multiplying each outcome by its probability t, overall, it's a valuable tool for assessing risks and aiding decision-making processes.