Problem 32 - Calculus for Biology and Medicine

Question

Assume that \(X\) is a discrete random variable with finite range. Show that if \(\operatorname{var}(X)=0\), then \(P(X=E(X))=1\)

Step-by-Step Solution

Answer

If \(\operatorname{var}(X) = 0\), then \(X\) must equal \(E(X)\) with probability 1, meaning \(P(X=E(X))=1\).

Step 1: Define Variance and Expectation

The variance of a random variable \(X\), \(\operatorname{var}(X)\), is defined as \( \operatorname{var}(X) = E[(X - E(X))^2] \). This measures the average squared deviation of \(X\) from its expected value \(E(X)\).

Step 2: Variance Equals Zero Condition

Given that \(\operatorname{var}(X) = 0\), we interpret this as the average of the squared deviations \((X - E(X))^2\) being zero. This implies that \(X - E(X) = 0\) almost surely, since non-zero terms would contribute to a positive variance.

Step 3: Interpret Probability Statement

Since \(X - E(X) = 0\), it follows that \(X = E(X)\) with probability 1. The probability \(P(X = E(X)) = 1\) concludes that \(X\) takes the constant value \(E(X)\) with absolute certainty.

Key Concepts

Understanding Variance

Variance is a key concept when analyzing discrete random variables. It quantifies how much the values of a random variable deviate from the mean, also known as the expected value. Variance is calculated using the formula \( \operatorname{var}(X) = E[(X - E(X))^2] \). This expression represents the mean of the squares of the tuple differences from the expected value.
To put it simply, think of variance as a measure of how 'spread out' the values of a variable are. When values are close to the expected value, the variance is smaller. Conversely, if values are widely scattered, the variance gets larger.
If the variance of a random variable \(X\) is zero, it specifically means there is no deviation at all. This implies every observation equals the expected value. Hence, \(X\) is a constant random variable in this case.

Exploring Expectation

Expectation is a fundamental concept in probability and statistics, often referred to as the "mean" or the "expected value". It symbolizes the average of all possible outcomes of a random variable, weighted by their probabilities. For a discrete random variable \(X\), the expectation is calculated as \( E(X) = \sum_{i} x_i P(X = x_i) \).
Expectation provides insight into the central tendency of the distribution of \(X\). It helps predict the average result of random phenomena over time. This makes expectation a powerful tool in various fields such as finance, insurance, and risk management.

• Expectation is essentially a weighted average of possible outcomes.
• A key use of expectation is to anticipate long-term averages.
• For discrete variables, each value is multiplied by its probability and then summed.

Deciphering Probability

Probability is the measure of the likelihood that a certain event will occur. When dealing with a discrete random variable, it's important to understand how probabilities associate with different outcomes. In simple terms, probability quantifies how likely a particular result is expected to be.
When the variance of a discrete random variable \(X\) is zero, as per the solution steps, it implies that the probability of \(X\) being equal to its expectation is 1, or 100%. This situation highlights the very nature of probability itself—certainty.

• Probability ranges between 0 and 1, where 0 indicates an impossible event and 1 denotes certainty.
• A discrete random variable with variance zero signifies that there is no other outcome than the expected one.
• Understanding probability allows for better decision-making under uncertainty.