Problem 4 - Calculus for Biology and Medicine

Question

Let \(X\) be a continuous random variable with density \(f(x)\), and assume that \(X \geq 2\). Why is \(E(X) \geq 2 ?\)

Step-by-Step Solution

Answer

Since all contributions to the expectation are from values \(\geq 2\), \(E(X) \geq 2\).

Step 1: Understanding the Expectation Definition

The expected value, or expectation, of a continuous random variable, denoted by \(E(X)\), is given by the integral \(E(X) = \int_{-\infty}^{\infty} x f(x) \, dx\), where \(f(x)\) is the probability density function of \(X\). It represents the average value or mean of \(X\) over all possible values weighted by their probabilities.

Step 2: Apply Condition on Random Variable

We know that \(X \geq 2\). This means the probability that \(X\) takes on values less than 2 is 0. Therefore, the density function \(f(x)\) is zero for all \(x < 2\).

Step 3: Simplify the Expected Value Expression

With the condition \(X \geq 2\), the expectation simplifies to \(E(X) = \int_{2}^{\infty} x f(x) \, dx\). Since the integrand \(x\) involves values \(x \geq 2\), the weight of \(x\) starts from 2 due to the integration bounds.

Step 4: Conclude Expectation is at Least 2

Since any value contributing to the expectation is \(\geq 2\), the mean or average of those values cannot be less than 2. Thus, the expectation \(E(X) \geq 2\) because we never take a weighted average of values less than 2.

Key Concepts

Expected Value

The concept of expected value in probability theory is crucial to understanding how we calculate the average outcome of a random variable. For any continuous random variable denoted as \(X\), its expected value \(E(X)\) is calculated using an integral:

• \(E(X) = \int_{-\infty}^{\infty} x f(x) \, dx\).
• Here, \(f(x)\) is the probability density function (PDF) of \(X\).

This formula represents the average or mean value of \(X\). What's fascinating about the expected value is how it accounts for all potential outcomes, each weighted by the likelihood of its occurrence, through the PDF.
In the provided exercise, we focused on a scenario where \(X \geq 2\). Since values lower than 2 have zero probability, the calculation simplifies to an integral starting from 2. This inherently makes the expected value at least 2, ensuring that the overall average cannot dip below this lower bound.
Understanding expectation helps you make informed predictions about random variables in various applications, from finance to physics.

Probability Density Function

A probability density function (PDF) is a fundamental concept defining how probabilities are distributed over the values of a continuous random variable. When dealing with continuous variables like our example with \(X\):

• \(f(x)\) describes the likelihood of \(X\) taking on a particular value \(x\).
• The integral of \(f(x)\) over an interval gives the probability that \(X\) falls within that interval.

Crucially, unlike discrete probabilities, \(f(x)\) doesn't give us probabilities directly, but rather probability densities. This requires using integration to determine the probability over an interval.
In practice, a PDF must satisfy two conditions:

• \(f(x) \geq 0\) for all \(x\), ensuring non-negative density.
• \(\int_{-\infty}^{\infty} f(x) \, dx = 1\), representing a total probability of 1 across the entire distribution.

In our example, because \(X \geq 2\), the PDF \(f(x)\) is zero for all \(x < 2\). This simplification ensures integration only happens from 2 onwards, affecting the expected value calculation directly.

Probability Theory

Probability theory is the branch of mathematics that deals with the analysis of random phenomena. It provides mathematical foundations to quantify uncertainty and predict outcomes in systems that exhibit randomness. Two central concepts are crucial:

• Random Variables: These are variables whose possible values are numerical outcomes of a random phenomenon. They can be either discrete or continuous.
• Probability Distributions: Describes how the probabilities are distributed over the values of the random variable.

For continuous random variables, probability distributions are expressed using probability density functions (PDFs), like \(f(x)\) in the exercise. This theory underlies the rationale for why expected values and PDFs are essential for understanding and predicting behaviors of random variables.
The intuition from probability theory is that when we have restrictions, such as \(X \geq 2\), we adjust our models accordingly, in this case by changing integration limits. It's a testament to the flexibility and robustness of probability theory in making sense of controlled randomness and structure in a chaotic world.