Problem 25 - Calculus for Biology and Medicine

Question

Assume that \(X\) is a discrete random variable with finite range, and set $$ p(x)=P(X=x) $$ (a) Show that $$ E(a X+b)=\sum_{x}(a x+b) p(x) $$ (b) Use your result in (a) and the rules for finite sums to conclude that $$ E(a X+b)=a E(X)+b $$

Step-by-Step Solution

Answer

(a) \( E(aX + b) = \sum_{x}(a x + b) p(x) \); (b) \( E(aX + b) = aE(X) + b \).

Step 1: Understanding the Expectation

The expectation, or expected value, of a discrete random variable \( X \) is calculated as \( E(X) = \sum_{x} x p(x) \), where \( p(x) = P(X = x) \). Thus, the expectation is a weighted average of possible values of \( X \).

Step 2: Substituting into the Expectation Formula

For part (a), we need to calculate the expectation of \( aX + b \). According to the definition, this expectation is:\[ E(aX + b) = \sum_{x} (a x + b) p(x) \]

Step 3: Apply Distribution of the Sum Inside the Expectation

We can break down the sum inside the expectation as:\[ E(aX + b) = \sum_{x} a x p(x) + \sum_{x} b p(x) \]This separates the constant terms from those related to \( X \).

Step 4: Factor Constants Out

Using the properties of sums, factor out the constant \( a \) from the first part and \( b \) from the second part:\[ E(aX + b) = a \sum_{x} x p(x) + b \sum_{x} p(x) \]Remember that \( \sum_{x} p(x) = 1 \) because it represents the probabilities of all possible outcomes of \( X \).

Step 5: Simplify the Expression

The first term becomes \( a E(X) \) and the second term simplifies to \( b \) because the sum of probabilities is 1:\[ E(aX + b) = a E(X) + b \]

Step 6: Conclusion

Therefore, using the expectation and rules for sums, we showed that:\[ E(aX + b) = aE(X) + b \] This proves both part (a) and part (b) as requested.

Key Concepts

Understanding Expectation in Discrete Random Variables

The expectation, also known as the expected value, is a fundamental concept when dealing with discrete random variables. It provides a measure of the central tendency of the possible values that the random variable can take. For a discrete random variable \(X\), with a range of possible outcomes and their corresponding probabilities, the expectation \(E(X)\) is expressed as: \[E(X) = \sum_{x} x p(x)\] Here's what happens in this formula:

• \(x\) represents each possible outcome of \(X\),
• \(p(x)\) represents the probability of \(X\) taking the value \(x\),
• The sum \(\sum_{x}\) indicates you're adding over all possible values of \(X\).

This formula can be thought of as a weighted average, where each outcome \(x\) is weighted by how likely it is to occur. The expectation provides insights into the overall behavior of the random variable. Think of it as the "average" outcome you'd expect if the random process was repeated many times. Understanding expectation helps in making predictions and decisions based on never-before-seen outcomes. It’s a key tool in both simple and complex probabilistic analyses. By becoming familiar with this concept, you can tackle many problems involving random variables more easily.

Probability and Its Role in Expectations

Probability serves as the foundation of the entire concept of expectation for discrete random variables. It's the mechanism that assigns likelihoods to each outcome. When dealing with probability, we follow a straightforward principle: the sum of probabilities across all possible outcomes must equal 1. This is owed to the fact that the probabilities encompass all potential scenarios.
For example, if you have a discrete random variable \(X\) that can take on values \(x_1, x_2, ..., x_n\), associated with probabilities \(p(x_1), p(x_2), ..., p(x_n)\), these probabilities obey: \[ \sum_{x} p(x) = 1 \]

• This ensures that the model of our random variable accounts for all possible outcomes.
• The individual probability \(p(x)\), being a part of this sum, impacts the expectation directly. More probable outcomes sway the expectation more heavily than less probable ones.

In simpler terms, probability enables us to weigh each outcome appropriately in the expectation calculation. The larger the probability of an outcome, the more influence that outcome has in determining the expectation. This probabilistic weighting helps in drawing more meaningful predictions and conclusions from data.

Finite Range of Discrete Random Variables

Discrete random variables are characterized by their finite range, meaning they can only take on a specific limited set of values. This is an important distinction from continuous random variables, which can take any value within a given range. The finiteness of the range simplifies computations, especially when dealing with expectations and probabilities.
A finite range means that there is a countable list of potential outcomes. Let's denote these outcomes as \(x_1, x_2, ..., x_n\). This finiteness allows us to precisely apply probabilities to each potential outcome, simplifying calculations and ensuring clarity in the analysis. With this set structure, calculations like expectation are straightforward, involving only a finite sum: \[E(X) = \sum_{x} x p(x)\]
This finite characteristic is particularly useful:

• It makes no need for complex integration, unlike continuous distributions.
• All possible values and their probabilities can be listed clearly, easing understanding and application.

Finite range brings predictability and simplicity, allowing for more direct assessments of random processes defined by the discrete random variable \(X\). This clarity extends to practical applications such as risk analysis, economic modeling, and game theory, where predicting outcomes based on discrete possibilities is essential.