Expected Values

The expected value or the population mean of a random variable indicates its central or average value. It is an important summary value of the distribution of the variable. In this article, we will look at the expected value of a random variable along with its uses and applications.

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Introduction to Expected Value

An expected value gives a quick insight into the behavior of a random variable without knowing if it is discrete or continuous.

Therefore, two random variables with the same expected value can have different probability distributions. Other descriptive measures like standard deviation also affect the shape of the distribution. The expected value of a random variable X is denoted as E(X).

Let’s say that X is a discrete random variable with possible values x₁, x₂, x₃, … , x_n. Also, p(x_i) denotes P(X = x_i). Therefore, the expected value of X is:

μ = E(X) = Σp(x_i) – x_i … where the elements are summed over all the values of the random variable X.

If X is a continuous random variable with a probability density function f(x), the the expected value of X is:

μ = E(X) = ∫x.f(x)dx

Learn more about Present Value here in detail.

Example of a Discrete Variable

When you throw a dice, each of the possible faces 1, 2, 3, 4, 5, 6(or the x_i‘s) has a probability of showing of \( \frac {1}{6} \) (the p(x_i)’s). Therefore, the expected value of the face showing is:

μ = E(X) = (1 x \( \frac {1}{6} \)) + (2 x \( \frac {1}{6} \)) + (3 x \( \frac {1}{6} \)) + (4 x \( \frac {1}{6} \)) + (5 x \( \frac {1}{6} \)) + (6 x \( \frac {1}{6} \)) = 3.5

Note that in this case, E(X) = 3.5 is not a possible value of X.

Expected Value of a Random Variable

In order to calculate the mean of a random variable, we do not simply add up the different variables. The expected value of a random variable is its mean. This value is the one you expect to obtain if you conduct an experiment whose outcomes are represented by the random variable.

In Probability Theory, the expected value or expectation or mathematical expectation or EV or mean refers to the value of a random variable that you expect if you repeat the random variable process infinite times and take an average of the obtained values.

In other words, an expected value is the weighted average of all possible values. The weights used in computing this average are the probabilities (for discrete random variables) or the values of a probability density function (for continuous random variables).

Example

A local club plans to invest Rs. 10,000 to host a football game. Further, it expects to sell tickets worth Rs. 15,000. However, if it rains on the day of the game, then it won’t sell any tickets and lose all the invested money. According to the weather forecast, there is a 20% possibility of rain on the day of the game. Is this a good investment?

Let’s make a table of probability distribution:

Outcome	+Rs. 5,000	-Rs. 10,000
Probability	0.80	0.20

Therefore, the expected value is

E(X) = 5000 x 0.8 – 10000 x 0.2 = 4000 – 2000 = Rs. 2,000

Hence, the club can expect a return of Rs. 2,000.

Illustration

A company manufactures electronic gadgets. 1 out of every 50 gadgets is faulty. However, the company does not know which ones are faulty until they receive a complaint from the buyer. The company makes a profit of Rs.30 on the sale of a working gadget but suffers a loss of Rs.800 for every faulty gadget. Can the company make a profit in the long term?

Solution

Let’s find the expected value:

E(X) = 30 x \( \frac {49}{50} \) – 800 x \( \frac {1}{50} \) = \( \frac {1470}{50} \) – \( \frac {800}{50} \) = \( \frac {670}{50} \) = Rs.13.4

Therefore, the company can make a profit of Rs.13.4 per gadget produced.

We can interpret the expected value as the long-run average of results of many independent repetitions of an experiment. Let’s say that a random variable X can take a value x₁ with a probability p₁, value x₂ with a probability p₂, and so on, up to value x_k with a probability p_k. The expected value of this random variable is:

E(X) = x₁p₁ + x₂p₂ + … + x_kp_k

Since all probabilities p_i add up to 1 (p₁ + p₂ + … p_k = 1), the expected value is the weighted average with p_i‘s being the weights:

E(X) = \( \frac {x_1p_1 + x_2p_2 + … + x_kp_k}{1} \) = \( \frac {x_1p_1 + x_2p_2 + … + x_kp_k}{p_1 + p_2 + … + p_k} \)

If all outcomes are equally likely, that is, p₁ = p₂ = … = p_k, then the weighted average becomes a simple average. Therefore, since the expected value of a random variable is the aveerage of all possible values, this is the value that we expect on an average.

If all the outcomes are not equally probable, then the simple average is replaced with the weighted average.

Uses and Applications

The expected value has an important role to play in many applications. When we do regression analysis, we desire a formula for the observed data which gives a good estimate of the parameter giving the effect of some explanatory variables upon a dependent variable.

This formula also gives different estimates using different samples of data. Therefore, the estimate itself gives a random variable. Hence, we consider a formula if it is an unbiased estimator or the estimated value of the estimate is equal to the true value of the desired parameter.

On the other hand, in decision theory, particularly in the choice under uncertainty, we describe an agent as making an optimal choice in the context of incomplete information.

For the risk-neutral agents, the choice involves using the expected values of uncertain quantities. For the risk-averse agents, on the other hand, it involves maximizing the expected value of some objective function.

Solved Question

Q1. What is the expected value of a random variable?

Answer: The expected value of a random variable is its mean. It is the value you expect to obtain if you conduct an experiment whose outcomes are represented by the random variable. It is the weighted average of all possible values where the probabilities of occurrence of the values are the weights.